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Marat V. Markin Elementary Functional Analysis Also of Interest Real Analysis. Measure and Integration Marat V. Markin, 2019 ISBN 9783110600971, eISBN (PDF) 9783110600995, eISBN (EPUB) 9783110598827 Elementary Operator Theory Marat V. Markin, 2019 ISBN 9783110600964, eISBN (PDF) 9783110600988, eISBN (EPUB) 9783110598889 Applied Nonlinear Functional Analysis. An Introduction Nikolaos S. Papageorgiou, Patrick Winkert, 2018 ISBN 9783110516227, eISBN (PDF) 9783110532982, eISBN (EPUB) 9783110531831 Functional Analysis. A Terse Introduction Gerardo Chacón, Humberto Rafeiro, Juan Camilo Vallejo, 2017 ISBN 9783110441918, eISBN (PDF) 9783110441925, eISBN (EPUB) 9783110433647 Complex Analysis. A Functional Analytic Approach Friedrich Haslinger, 2017 ISBN 9783110417234, eISBN (PDF) 9783110417241, eISBN (EPUB) 9783110426151 Marat V. Markin Elementary Functional Analysis  Mathematics Subject Classification 2010 4602, 4702, 46A22, 46A30, 46A32, 46A35, 46A45, 46B03, 46B04, 46B10, 46C05, 47A30 Author Prof. Dr. Marat V. Markin California State University, Fresno Department of Mathematics 5245 North Backer Avenue, M/S PB 108 Fresno, CA 93740 USA mmarkin@csufresno.edu ISBN 9783110613919 eISBN (PDF) 9783110614039 eISBN (EPUB) 9783110614091 Library of Congress Control Number: 2018950580 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2018 Walter de Gruyter GmbH, Berlin/Boston Cover image: Mordolff / Getty Images Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books Lecks, GmbH www.degruyter.com  To my students, past, present, and future. Preface Functional analysis arose in the early twentieth century and gradually, conquering one stronghold after another, became a nearly universal mathematical doctrine, not merely a ; new area of mathematics, but a new mathematical world view. Its appearance was the inevitable consequence of the evolution of all of nineteenthcentury mathematics, in particular classical analysis and mathematical physics. Its original basis was formed by Cantor’s theory of sets and linear algebra. Its existence answered the question of how to state general principles of a broadly interpreted analysis in a way suitable for the most diverse situations. A. M. Vershik Having at once abandoned futile attempts to say anything better to describe the essence and origins of functional analysis than the above epigraph, the author, echoing [32], could not but choose it as a foreword for this book. And yet, a few more words are to be uttered. Functional analysis The emergence of functional analysis, a vast and rapidly growing branch of modern mathematics using “the intuition and language of geometry in the study of functions” [54], was brought to life by the inherent to mathematics epistemological tendency towards unification and abstraction. The constantly widening field of applications of functional analysis encompasses ordinary and partial differential equations, numerical analysis, calculus of variations, approximation theory, integral equations, and much more. The profoundly abstract nature and extensive applicability of functional analysis make a course in it to be an indispensable part of a contemporary graduate mathematics curriculum increasing its value not only for graduate students majoring in mathematics, but also for those majoring in physics, science, and engineering. The purpose of the book and targeted audience While there is a plethora of excellent, but mostly “tellitall” books on the subject (see, e. g., a rather extensive bibliography below), this one is intended to take a unique place in what today seems to be a still wide open niche for an introductory text on the basics of functional analysis to be taught within the existing constraints of the standard, for the United States, onesemester graduate curriculum (fifteen weeks with two seventyfiveminute lectures per week). The prerequisites are set intentionally quite low, the students not being assumed to have taken graduate courses in real or complex analysis and general topology, to make the course accessible and attractive to a wider audience of STEM (science, technology, engineerhttps://doi.org/10.1515/9783110614039201 VIII  Preface ing, and mathematics) graduate students or advanced undergraduates with a solid background in calculus and linear algebra. This is why the examples are primarily built around sequence spaces, L2 spaces being mentioned only tangentially whenever pertinent. Book’s scope and specifics The book consists of seven chapters and an appendix taking the reader from the fundamentals of abstract spaces (metric, vector, normed vector, and inner product), through the basics of linear operators and functionals, the three fundamental principles (the HahnBanach Theorem (the extension form), the Uniform Boundedness Principle, the Open Mapping Theorem and its equivalents: the Inverse Mapping and Closed Graph Theorems) with their numerous profound implications and certain interesting applications, to the elements of the duality and reflexivity theory. Chapter 1 outlines some necessary preliminaries, while the Appendix gives a concise discourse on the celebrated Axiom of Choice, its equivalents (the Hausdorff Maximal Principle, Zorn’s Lemma, and Zermello’s WellOrdering Principle), and ordered sets. Being designed as a text to be used in a classroom, the book constantly calls for the student’s actively mastering the knowledge of the subject matter. It contains 112 Problems, which are indispensable for understanding and moving forward. Many important statements, such as Cantor’s Intersection Theorem (Theorem 2.55), are given as problems, a lot of these are frequently referred to and used in the main body. There are also 376 Exercises throughout the text, including Chapter 1 and the Appendix, which require of the student to prove or verify a statement or an example, fill in necessary details in a proof, or provide an intermediate step or a counterexample. They are also an inherent part of the material. More difficult problems, such as Section 2.19, Problems 6 and 7, are marked with an asterisk, many problem and exercises being supplied with “existential” hints. The book is generous on Examples and contains numerous Remarks accompanying every definition and virtually each statement to discuss certain subtleties, raise questions on whether the converse assertions are true, whenever appropriate, or whether the conditions are essential. As amply demonstrated by experience, students tend to better remember statements by their names rather than by numbers. Thus, a distinctive feature of this book is that every theorem, proposition, and corollary, unless already possessing a name, is endowed with a descriptive one, making it easier to remember, which, in this author’s humble opinion, is quite a bargain when the price for better understanding and retention of the material is a little clumsiness while making a longer reference. Each statement is referred to by its name and not just the number. e. g., the Representation Theorem for lp∗ (1 ≤ p < ∞) (Theorem 7.3), as opposed to merely Theorem 7.3. Acknowledgments  IX Acknowledgments I am eternally grateful, to my mother, Svetlana A. Markina, for her unfailing faith, support, and endless patience, without which this and many other endeavors of mine would be impossible. My utmost appreciation goes to Mr. Edward Sichel, my pupil and graduate advisee, for his invaluable assistance with proofreading and improving the manuscript. I am also very thankful to Dr. Przemyslaw Kajetanowicz (Department of Mathematics, CSU, Fresno) for his kind assistance with some figures. My sincere acknowledgments are also due to the following associates of the Walter de Gruyter GmbH: Dr. Apostolos Damialis, Acquisitions Editor in Mathematics, for seeing value in my manuscript and making authors his highest priority, Ms. Nadja Schedensack, Project Editor in Mathematics and Physics, for her superb efficiency in managing all project related matters, as well as Ms. Ieva Spudulytė and Ms. Ina Talandienė, VTeX Book Production, for their expert editorial and LATEX typesetting contributions. Clovis, California June 2018 Marat V. Markin Contents Preface  VII 1 1.1 1.1.1 1.1.2 1.2 1.3 Preliminaries  1 Set Theoretic Basics  1 Some Terminology and Notations  1 Cardinality and Countability  2 Terminology Related to Functions  4 Upper and Lower Limits  6 2 Metric Spaces  7 2.1 Definition and Examples  7 2.2 Hölder’s and Minkowski’s Inequalities  9 2.2.1 Conjugate Indices  9 2.2.2 Young’s Inequality  9 2.2.3 The Case of nTuples  10 2.2.4 Sequential Case  12 2.3 Subspaces of a Metric Space  14 2.4 Function Spaces  14 2.5 Further Properties of Metric  16 2.6 Convergence and Continuity  17 2.6.1 Convergence of a Sequence  17 2.6.2 Continuity, Uniform Continuity, and Lipschitz Continuity  18 2.7 Balls, Separation, and Boundedness  20 2.8 Interior Points, Open Sets  23 2.9 Limit Points, Closed Sets  24 2.10 Dense Sets and Separable Spaces  27 2.11 Exterior and Boundary  28 2.12 Equivalent Metrics, Homeomorphisms and Isometries  29 2.12.1 Equivalent Metrics  29 2.12.2 Homeomorphisms and Isometries  30 2.13 Completeness and Completion  31 2.13.1 Cauchy/Fundamental Sequences  31 2.13.2 Complete Metric Spaces  33 2.13.3 Subspaces of Complete Metric Spaces  37 2.13.4 Nested Balls Theorem  37 2.13.5 Completion  40 2.14 Category and Baire Category Theorem  43 2.14.1 Nowhere Denseness  43 2.14.2 Category  46 XII  Contents 2.14.3 2.15 2.15.1 2.15.2 2.15.3 2.15.4 2.15.5 2.15.6 2.15.7 2.16 2.17 2.17.1 2.17.2 2.17.3 2.18 2.18.1 2.18.2 2.18.3 2.18.4 2.19 3 3.1 3.1.1 3.1.2 3.1.3 3.1.4 3.1.5 3.1.6 3.1.7 3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5 3.2.6 3.3 3.3.1 3.3.2 3.4 Baire Category Theorem  47 Compactness  49 Total Boundedness  50 Compactness, Precompactness  54 Hausdorff Criterion  60 Compactness in Certain Complete Metric Spaces  62 Other Forms of Compactness  64 Equivalence of Different Forms of Compactness  65 Compactness and Continuity  67 Space (C(X , Y ), ρ∞ )  70 Arzelà–Ascoli Theorem  72 Uniform Boundedness and Equicontinuity  72 Arzelà–Ascoli Theorem  73 Application: Peano’s Existence Theorem  76 Stone–Weierstrass Theorem  79 Weierstrass Approximation Theorem  79 Algebras  79 Stone–Weierstrass Theorem  82 Applications  87 Problems  87 Normed Vector and Banach Spaces  97 Vector Spaces  97 Definition, Examples, Properties  97 Homomorphisms and Isomorphisms  100 Subspaces  101 Spans and Linear Combinations  103 Linear Independence, Hamel Bases, Dimension  103 New Spaces from Old  108 Disjoint and Complementary Subspaces, Direct Sum Decompositions, Deficiency and Codimension  111 Normed Vector and Banach Spaces  114 Definitions and Examples  114 Series and Completeness Characterization  118 Comparing Norms, Equivalent Norms  119 Isometric Isomorphisms  120 Completion  121 Topological and Schauder Bases  122 FiniteDimensional Spaces and Related Topics  125 Norm Equivalence and Completeness  125 FiniteDimensional Subspaces and Bases of Banach Spaces  127 Riesz’s Lemma and Implications  130 Contents  XIII 3.5 3.5.1 3.5.2 3.6 Convexity, Strictly Convex Normed Vector Spaces  132 Convexity  132 Strictly Convex Normed Vector Spaces  133 Problems  137 4 Inner Product and Hilbert Spaces  141 4.1 Definitions and Examples  141 4.2 Inner Product Norm, Cauchy–Schwarz Inequality  143 4.3 Hilbert Spaces  145 4.4 Certain Geometric Properties  147 4.4.1 Polarization Identities  147 4.4.2 Parallelogram Law  147 4.4.3 Orthogonality  151 4.5 Nearest Point Property  152 4.6 Projection Theorem  155 4.6.1 Orthogonal Complements  155 4.6.2 Projection Theorem  157 4.7 Completion  159 4.8 Gram Determinant  160 4.9 Orthogonal and Orthonormal Sets  165 4.10 Gram–Schmidt Process  170 4.11 Generalized Fourier Series  171 4.11.1 Finite Orthonormal Set  172 4.11.2 Arbitrary Orthonormal Set  174 4.11.3 Orthonormal Sequence  178 4.12 Orthonormal Bases and Orthogonal Dimension  179 4.13 Problems  184 5 5.1 5.1.1 5.1.2 5.1.3 5.2 5.2.1 5.2.2 5.3 5.4 Linear Operators and Functionals  187 Linear Operators and Functionals  187 Definitions and Examples  187 Kernel, Range, and Graph  189 RankNullity and Extension Theorems  189 Bounded Linear Operators and Functionals  192 Definitions, Properties, and Examples  192 Space of Bounded Linear Operators, Dual Space  197 Closed Linear Operators  201 Problems  204 6 6.1 6.1.1 Three Fundamental Principles of Linear Functional Analysis  207 Hahn–Banach Theorem  207 Hahn–Banach Theorem for Real Vector Spaces  207 XIV  Contents 6.1.2 6.2 6.2.1 6.2.2 6.2.3 6.2.4 6.2.5 6.2.6 6.3 6.4 6.4.1 6.4.2 6.5 6.5.1 6.5.2 6.6 6.6.1 6.6.2 6.6.3 6.6.4 6.7 7 7.1 7.1.1 7.1.2 7.1.3 7.2 7.2.1 7.2.2 7.3 7.3.1 7.3.2 7.3.3 7.4 7.4.1 7.4.2 7.5 7.5.1 7.5.2 7.5.3 Hahn–Banach Theorem for Normed Vector Spaces  210 Implications of the Hahn–Banach Theorem  212 Separation and Norm Realization  212 Characterization of Fundamentality  215 Sufficiency for Separability  216 Isometric Embedding Theorems  217 Second Dual Space and Canonical Isomorphism  218 Closed Complemented Subspaces  219 Weak and Weak* Convergence  221 Uniform Boundedness Principle, the Banach–Steinhaus Theorem  225 Uniform Boundedness Principle  225 Banach–Steinhaus Theorem  228 Applications of the Uniform Boundedness Principle  231 Weak Boundedness  232 Matrix Methods of Convergence and Summability  233 Open Mapping, Inverse Mapping, and Closed Graph Theorems  240 Open Mapping Theorem  240 Inverse Mapping Theorem and Applications  245 Closed Graph Theorem and Application  249 Equivalence of OMT, IMT, and CGT  255 Problems  256 Duality and Reflexivity  261 SelfDuality of Hilbert Spaces  261 Riesz Representation Theorem  261 Linear Bounded Functionals on Certain Hilbert Spaces  264 Weak Convergence in Hilbert Spaces  264 Duality of FiniteDimensional Spaces  265 Representation Theorem  265 Weak Convergence in FiniteDimensional Spaces  268 Duality of Sequence Spaces  268 Representation Theorem for lp∗  268 Weak Convergence in lp (1 ≤ p < ∞)  272 Duality and Weak Convergence for (c0 , ‖ ⋅ ‖∞ )  276 Duality and Weak Convergence for (C[a, b], ‖ ⋅ ‖∞ )  277 Riesz Representation Theorem for C ∗ [a, b]  277 Weak Convergence in (C[a, b], ‖ ⋅ ‖∞ )  278 Reflexivity  278 Definition and Examples  278 Completeness of a Reflexive Space  283 Reflexivity of a Closed Subspace  283 Contents  XV 7.5.4 7.5.5 7.5.6 7.5.7 7.6 A A.1 A.1.1 A.1.2 A.1.3 A.2 A.3 Isometric Isomorphism and Reflexivity  285 Characterization of Reflexivity  286 Weak Convergence and Weak Completeness  287 Bounded Sequence Property  288 Problems  290 The Axiom of Choice and Equivalents  293 The Axiom of Choice  293 The Axiom of Choice  293 Controversy  293 Timeline  294 Ordered Sets  294 Equivalents  298 Bibliography  303 Index  305 1 Preliminaries In this chapter, we outline certain terminology, notations, and preliminary facts essential to our subsequent discourse. 1.1 Set Theoretic Basics 1.1.1 Some Terminology and Notations – – – – – – – – – The logic quantifiers ∀, ∃, and ∃! stand for “for all”, “there exist(s)”, and “there exists a unique”, respectively. ℕ := {1, 2, 3, . . . } is the set of natural numbers. ℤ := {0, ±1, ±2, . . . } is the set of integers. ℚ is the set of rational numbers. ℝ is the set of real numbers. ℂ is the set of complex numbers. ℤ+ , ℚ+ , and ℝ+ are the sets of nonnegative integers, rationals, and reals, respectively. ℝ := [−∞, ∞] is the set of extended real numbers (extended real line). For n ∈ ℕ, ℝn and ℂn are the nspaces of all ordered ntuples of real and complex numbers, respectively. Let X be a set. Henceforth, all sets are supposed to be subsets of X. – P (X) is the power set of X, i. e., the collection of all subsets of X. – 2X is the set of all binary functions f : X → {0, 1}, provided X ≠ 0. – Sets A, B ⊆ X with A ∩ B = 0 are called disjoint. – Let I be a nonempty indexing set. The sets of a collection {Ai }i∈I of subsets of X are said to be pairwise disjoint if Ai ∩ Aj = 0, – – i, j ∈ I, i ≠ j. For A, B ⊆ X, A \ B := {x ∈ X  x ∈ A, but x ∉ B} is the difference of A and B, in particular, Ac := X \ A = {x ∈ X  x ∉ A} is the complement of A and A \ B = A ∩ Bc ; Let I be a nonempty indexing set and {Ai }i∈I be a collection of subsets of X. De Morgan’s laws state c (⋃ Ai ) = ⋂ Aci i∈I i∈I and c (⋂ Ai ) = ⋃ Aci . i∈I i∈I More generally, B \ ⋃ Ai = ⋂ B \ Ai i∈I i∈I https://doi.org/10.1515/9783110614039001 and B \ ⋂ Ai = ⋃ B \ Ai . i∈I i∈I 2  1 Preliminaries – The Cartesian product of sets Ai ⊆ X, i = 1, . . . , n (n ∈ ℕ), A1 × ⋅ ⋅ ⋅ × An := {(x1 , . . . , xn )  xi ∈ Ai , i = 1, . . . , n} . 1.1.2 Cardinality and Countability Definition 1.1 (Similarity of Sets). Sets A and B are said to be similar if there exists a onetoone correspondence (bijection) between them. Notation. A ∼ B. Remark 1.1. Similarity is an equivalence relation (reflexive, symmetric, and transitive) on the power set P (X) of a nonempty set X. Exercise 1.1. Verify. Thus, in the contualtext, we can use the term “equivalence” synonymously to “similarity”. Definition 1.2 (Cardinality). Equivalent sets are said to have the same number of elements or cardinality. Cardinality is a characteristic of an equivalence class of similar sets. Notation. P (X) ∋ A → A. Remark 1.2. Thus, A ∼ B iff A = B, i. e., two sets are equivalent iff they share the same cardinality. Examples 1.1. 1. For a nonempty set X, P (X) ∼ 2X . 2. ℕ = ℤ = ℚ := ℵ0 . 3. [0, 1] = ℝ = ℂ := c. See, e. g., [18, 21]. Definition 1.3 (Domination). If sets A and B are such that A is equivalent to a subset of B, we write A⪯B and say that B dominates A. If, in addition, A ≁ B, we write A≺B and say that B strictly dominates A. Remark 1.3. The relation ⪯ is a partial order (reflexive, antisymmetric, and transitive) on the power set P (X) of a nonempty set X (see Appendix A). 1.1 Set Theoretic Basics  3 Exercise 1.2. Verify reflexivity and transitivity. The antisymmetry of ⪯ is the subject of the following celebrated theorem. Theorem 1.1 (Schröder–Bernstein Theorem). If, for sets A and B, A ⪯ B and B ⪯ A, then A ∼ B.1 For a proof, see, e. g., [18]. Remark 1.4. The set partial order ⪯ defines a partial order ≤ on the set of cardinals: A ≤ B ⇔ A ⪯ B. Thus, the SchröderBernstein Theorem can be equivalently reformulated in terms of the cardinalities as follows: If, for sets A and B, A ≤ B and B ≤ A, then A = B. Theorem 1.2 (Cantor’s Theorem). Every set X is strictly dominated by its power set P (X):2 X ≺ P (X). Equivalently, X < P (X). For a proof, see, e. g., [18]. In view of Examples 1.1, we obtain the following Corollary 1.1. For a nonempty set X, X ≺ 2X , i. e., X < 2X . Definition 1.4 (Countable/Uncountable Set). A countable set is a set with the same cardinality as a subset of the set ℕ of natural numbers, i. e., equivalent to a subset of ℕ. A set that is not countable is called uncountable. Remarks 1.5. – A countable set A is either finite, i. e., equivalent to a set of the form {1, . . . , n} ⊂ ℕ with some n ∈ ℕ, in which case, we say that A has n elements, or countably infinite, i. e., equivalent to the entire ℕ. – For a finite set A of n elements (n ∈ ℕ), A = {1, . . . , n} = n, 1 Ernst Schröder (1841–1902), Felix Bernstein (1878–1956). 2 Georg Cantor (1845–1918). 4  1 Preliminaries For a countably infinite set A, A = ℕ = ℵ0 – (see Examples 1.1). In some sources, the term “countable” is used in the sense of “countably infinite”. To avoid ambiguity, the term “at most countable” can be used when finite sets are included in the consideration. The subsequent statement immediately follows from Cantor’s Theorem (Theorem 1.2). Proposition 1.1 (Uncountable Sets). The sets P (ℕ) and 2ℕ (the set of all binary sequences) are uncountable. Theorem 1.3 (Properties of Countable Sets). (1) Every infinite set contains a countably infinite subset (based on the Axiom of Choice (see Appendix A)). (2) Any subset of a countable set is countable. (3) The union of countably many countable sets is countable. (4) The Cartesian product of finitely many countable sets is countable. Exercise 1.3. Prove that (a) the set ℤ of all integers and the set of all rational numbers are countable; (b) for any n ∈ ℕ, ℤn and ℚn are countable; (c) the set of all algebraic numbers (the roots of polynomials with integer coefficients) is countable. Subsequently, we also need the following useful result. Proposition 1.2 (Cardinality of the Collection of Finite Subsets). The cardinality of the collection of all finite subsets of an infinite set coincides with the cardinality of the set. For a proof, see, e. g., [21, 26, 38]. 1.2 Terminology Related to Functions Let X and Y be nonempty sets and 0 ≠ D ⊆ X f : D → Y. – – The set D is called the domain (of definition) of f . The value of f corresponding to an x ∈ D is designated by f (x). 1.2 Terminology Related to Functions  5 – The set {f (x)  x ∈ D} – of all values of f is called the range of f (also the codomain or target set). For a set A ⊆ D, the set of values of f corresponding to all elements of A f (A) := {f (x)  x ∈ A} – is called the image of A under the function f . Thus, the range of f is the image f (D) of the whole domain D. For a set B ⊆ Y, the set of all elements of the domain that map to the elements of B f −1 (B) := {x ∈ D  f (x) ∈ B} is called the inverse image (or preimage) of B. Example 1.2. For X = Y := ℝ, f (x) := x2 , and D := [−1, 2], – f ([−1, 2]) = f ([1, 2]) = [1, 4]. – f −1 ([−2, −1]) = 0, f −1 ([0, 1]) = [−1, 1], f −1 ([1, 4]) = [−1, 2]. Theorem 1.4 (Properties of Inverse Image). Let X and Y be nonempty sets and 0 ≠ D ⊆ X f : D → Y. Then, for an arbitrary nonempty collection {Bi }i∈I of subsets of Y, (1) f −1 (⋃i∈I Bi ) = ⋃i∈I f −1 (Bi ), (2) f −1 (⋂i∈I Bi ) = ⋂i∈I f −1 (Bi ), and (3) for any B1 , B2 ⊆ Y, f −1 (B1 \ B2 ) = f −1 (B1 ) \ f −1 (B2 ), i. e., preimage preserves all set operations. Exercise 1.4. (a) Prove. (b) Show that image preserves unions, i. e., for an arbitrary nonempty collection {Ai }i∈I of subsets of D, f (⋃ Ai ) = ⋃ f (Ai ), i∈I i∈I and unions only. Give corresponding counterexamples for intersections and differences. 6  1 Preliminaries 1.3 Upper and Lower Limits Definition 1.5 (Upper and Lower Limits). Let (xn )n∈ℕ (another notation is {xn }∞ n=1 ) be a sequence of real numbers. The upper limit or limit superior of (xn )n∈ℕ is defined as follows: lim x n→∞ n := lim sup xk = inf sup xk ∈ [−∞, ∞]. n→∞ k≥n n∈ℕ k≥n The lower limit or limit inferior of {xn }∞ n=1 is defined as follows: lim xn := lim inf xk = sup inf xk ∈ [−∞, ∞]. n→∞ n→∞ k≥n n∈ℕ k≥n Alternative notations are lim supn→∞ xn and lim infn→∞ xn , respectively. Example 1.3. For n, n is odd, −1/n, n is even, xn := { lim x n→∞ n n ∈ ℕ, = ∞ and lim xn = 0. n→∞ Exercise 1.5. (a) Verify. (b) Explain why the upper and lower limits, unlike the regular limit, are guaranteed to exist for an arbitrary sequence of real numbers. (c) Show that lim xn ≤ lim xn . n→∞ n→∞ Proposition 1.3 (Characterization of Limit Existence). For a sequence of real numbers {xn }∞ n=1 , lim x n→∞ n ∈ [−∞, ∞] exists iff lim xn = lim xn , n→∞ n→∞ in which case lim x n→∞ n = lim xn = lim xn . n→∞ n→∞ 2 Metric Spaces In this chapter, we study abstract sets endowed with a notion of distance, whose properties mimic those of the regular distance in threedimensional space. Distance brings to life various topological notions such as limit, continuity, openness, closedness, compactness, denseness, and the geometric notion of boundedness, as well as the notions of fundamentality of sequences and completeness. We consider all these and more in depth here. 2.1 Definition and Examples Definition 2.1 (Metric Space). A metric space is a nonempty set X with a metric (or distance function), i. e., a mapping ρ(⋅, ⋅) : X × X → ℝ subject to the following metric axioms: 1. ρ(x, y) ≥ 0, x, y ∈ X. 2. ρ(x, y) = 0 iff x = y. 3. ρ(x, y) = ρ(y, x), x, y ∈ X. 4. ρ(x, z) ≤ ρ(x, y) + ρ(y, z), x, y, z ∈ X. Nonnegativity Separation Symmetry Triangle Inequality For any fixed x, y ∈ X, the number ρ(x, y) is called the distance of x from y, or from y to x, or between x and y. Notation. (X, ρ). Remark 2.1. A function ρ(⋅, ⋅) : X × X → ℝ satisfying the metric axioms of symmetry, triangle inequality, and the following weaker form of the separation axiom: 2w. ρ(x, y) = 0 if x = y also necessarily satisfies the axiom of nonnegativity and is called a semimetric (or pseudometric) on X (see the examples to follow). Exercise 2.1. Verify that 2w, 3, and 4 imply 1. Examples 2.1. 1. Any nonempty set X is a metric space relative to the discrete metric 0 X ∋ x, y → ρd (x, y) := { 1 2. if x = y, if x ≠ y. The real line ℝ or the complex plane ℂ is a metric space relative to the regular distance function ρ(x, y) := x − y. https://doi.org/10.1515/9783110614039002 8  2 Metric Spaces 3. Let n ∈ ℕ and 1 ≤ p ≤ ∞. The real/complex nspace ℝn or ℂn is a metric space relative to the pmetric 1/p [∑ni=1 xi − yi p ] ρp (x, y) = { max1≤i≤n xi − yi  if 1 ≤ p < ∞, if p = ∞, where x := (x1 , . . . , xn ) and y := (y1 , . . . , yn ), designated by lp(n) (real or complex, respectively). Remarks 2.2. – For n = 1, all these metrics coincide with ρ(x, y) = x − y. – For n = 2, 3, and p = 2, we have the usual Euclidean distance. – (ℂ, ρ) = (ℝ2 , ρ2 ). 4. Let 1 ≤ p ≤ ∞. The set lp of all real/complex sequences {xk }∞ k=1 satisfying ∞ ∑ xk p < ∞ (1 ≤ p < ∞), sup xk  < ∞ (p = ∞) k=1 k∈ℕ (psummable/bounded sequences, respectively) is a metric space relative to the pmetric 1/p p [∑∞ k=1 xk − yk  ] ρp (x, y) = { supk∈ℕ xk − yk  if 1 ≤ p < ∞, if p = ∞, ∞ where x := {xk }∞ k=1 , y := {yk }k=1 ∈ lp . Remark 2.3. When it is contextually important to distinguish between the real and complex cases, we use the notations lp(n) (ℝ), lp (ℝ) and lp(n) (ℂ), lp (ℂ), respectively. Exercise 2.2. Verify Examples 2.1, 1, 2 and 3, 4 for p = 1 and p = ∞. Remark 2.4. While verifying Examples 2.1, 3 and 4 for p = 1 and p = ∞ is straightforward, proving the triangle inequality for 1 < p < ∞ requires Minkowski’s1 inequality. 1 Hermann Minkowski (1864–1909). 2.2 Hölder’s and Minkowski’s Inequalities  9 2.2 Hölder’s and Minkowski’s Inequalities Here, we are to prove the celebrated Hölder’s2 and Minkowski’s inequalities for ntuples (n ∈ ℕ) and sequences. We use Hölder’s inequality to prove Minkowski’s inequality for the case of ntuples. In turn, to prove Hölder’s inequality for ntuples, we rely upon Young’s3 inequality, and hence, the latter is to be proved first. For the sequential case, Hölder’s and Minkowski’s inequalities are proved independently based on their analogues for ntuples. 2.2.1 Conjugate Indices Definition 2.2 (Conjugate Indices). We call 1 ≤ p, q ≤ ∞ conjugate indices if they are related as follows: 1 1 + = 1 for 1 < p, q < ∞, p q q=∞ for p = 1, q=1 for p = ∞. Examples 2.2. In particular, p = 2 and q = 2 are conjugate as well as p = 3 and q = 3/2. Remark 2.5. Thus, for 1 < p, q < ∞, q= ∞, p → 1+, 1 p =1+ →{ p−1 p−1 1, p → ∞, q > 2 if 1 < p < 2 and 1 < q < 2 if p > 2, and the following relationships hold: p + q = pq, pq − p − q + 1 = 1 ⇒ (p − 1)(q − 1) = 1, p (p − 1)q = p ⇒ p − 1 = , q q (q − 1)p = q ⇒ q − 1 = . p (2.1) 2.2.2 Young’s Inequality Theorem 2.1 (Young’s Inequality). Let 1 < p, q < ∞ be conjugate indices. Then, for any a, b ≥ 0, ab ≤ 2 Otto Ludwig Hölder (1859–1937). 3 William Henry Young (1863–1942). ap bq + . p q 10  2 Metric Spaces Proof. The inequality is, obviously, true if a = 0 or b = 0 and for p = q = 2. Exercise 2.3. Verify. Suppose that a, b > 0, 1 < p < 2 or p > 2 and recall that (p − 1)(q − 1) = 1 and (p − 1)q = p. (see (2.1)). Comparing the areas in Figure 2.1, which corresponds to the case of p > 2, the case of 1 < p < 2 being symmetric, we conclude that A ≤ A1 + A2 , where A is the area of the rectangle [0, a] × [0, b], the equality being the case iff b = ap−1 = ap/q . Hence, a ab ≤ ∫ x p−1 b dx + ∫ yq−1 dy = 0 0 ap bq + . p q Remark 2.6. As Figure 2.1 shows, equality in Young’s Inequality (Theorem 2.1) holds iff ap = bq . Figure 2.1: The case of 2 < p < ∞. 2.2.3 The Case of nTuples Definition 2.3 (pNorm of an nTuple). Let n ∈ ℕ. For an ntuple x := (x1 , . . . , xn ) ∈ ℂn (1 ≤ p ≤ ∞), the pnorm of x is the distance of x from the zero ntuple 0 := (0, . . . , 0) in lp(n) : 1/p [∑ni=1 xi p ] ‖x‖p := ρp (x, 0) = { max1≤i≤n xi  if 1 ≤ p < ∞, if p = ∞. 2.2 Hölder’s and Minkowski’s Inequalities  11 Remarks 2.7. – For an x := (x1 , . . . , xn ) ∈ ℂn , ‖x‖p = 0 ⇔ x = 0. – Observe that, for any x := (x1 , . . . , xn ), y := (y1 , . . . , yn ) ∈ ℂn , ρp (x, y) = ‖x − y‖p , where x − y = (x1 , . . . , xn ) − (y1 , . . . , yn ) := (x1 − y1 , . . . , xn − yn ). Exercise 2.4. Verify. Theorem 2.2 (Hölder’s Inequality for nTuples). Let n ∈ ℕ and 1 ≤ p, q ≤ ∞ be conjugate indices. Then, for any x := (x1 , . . . , xn ), y := (y1 , . . . , yn ) ∈ ℂn , n ∑ xi yi  ≤ ‖x‖p ‖y‖q . i=1 Proof. The symmetric cases of p = 1, q = ∞ and p = ∞, q = 1 are trivial. Exercise 2.5. Verify. Suppose that 1 < p, q < ∞ and let x := (x1 , . . . , xn ), y := (y1 , . . . , yn ) ∈ ℂn be arbitrary. If ‖x‖p = 0 or ‖y‖q = 0, which (see Remarks 2.7) is equivalent to x = 0 or y = 0, respectively, Hölder’s inequality is, obviously, true. If ‖x‖p ≠ 0 and ‖y‖q ≠ 0, applying Young’s Inequality (Theorem 2.1) to the nonnegative numbers aj = xj  ‖x‖p and bj = yj  ‖y‖q for each j = 1, . . . , n, we have: xj yj  ‖x‖p ‖y‖q ≤ p q 1 xj  1 yj  + , j = 1, . . . , n. p ∑ni=1 xi p q ∑ni=1 yi q We obtain Hölder’s inequality by adding the above n inequalities: ∑nj=1 xj yj  ‖x‖p ‖y‖q n p n q 1 ∑j=1 xj  1 ∑j=1 yj  1 1 ≤ + = + =1 p ∑ni=1 xi p q ∑ni=1 yi q p q and multiplying through by ‖x‖p ‖y‖q . 12  2 Metric Spaces Theorem 2.3 (Minkowski’s Inequality for nTuples). Let 1 ≤ p ≤ ∞. Then, for any (x1 , . . . , xn ), (y1 , . . . , yn ) ∈ ℂn , ‖x + y‖p ≤ ‖x‖p + ‖y‖p , where x + y = (x1 , . . . , xn ) + (y1 , . . . , yn ) := (x1 + y1 , . . . , xn + yn ). Proof. The cases of p = 1 or p = ∞ are trivial. Exercise 2.6. Verify. For an arbitrary 1 < p < ∞, we have: n n i=1 i=1 ∑ xi + yi p = ∑ xi + yi p−1 xi + yi  since xi + yi  ≤ xi  + yi , i = 1, . . . , n; n n i=1 i=1 ≤ ∑ xi + yi p−1 xi  + ∑ xi + yi p−1 yi  by Hölder’s inequality (Theorem 2.2); 1/q n ≤ [∑ xi + yi (p−1)q ] i=1 1/p n [∑ xi p ] i=1 1/q n + [∑ xi + yi (p−1)q ] i=1 n i=1 since 1/q n = [∑ xi + yi p ] i=1 1/q n = [∑ xi + yi p ] i=1 1/p n [∑ xi p ] i=1 n i=1 n + [∑ xi + yi p ] 1/p ([∑ xi p ] 1/q i=1 + [∑ yi p ] i=1 n 1/p (p − 1)q = p; [∑ yi p ] i=1 1/p n 1/p [∑ yi p ] ). Considering that, for ∑ni=1 xi + yi p = 0, Minkowski’s inequality trivially holds, suppose that ∑ni=1 xi + yi p > 0. Then, dividing through by [∑ni=1 xi + yi p ]1/q , in view of 1 − q1 = p1 , we see that, in this case, Minkowski’s inequality holds as well, which completes the proof. 2.2.4 Sequential Case Definition 2.4 (pNorm of a Sequence). For x := {xk }∞ k=1 ∈ lp (1 ≤ p ≤ ∞), the norm of x is the distance of x from the zero sequence 0 := {0, 0, 0, . . . } in lp : 1/p [∑∞ xk p ] ‖x‖p := ρp (x, 0) = { k=1 supk≥1 xk  if 1 ≤ p < ∞, if p = ∞. 2.2 Hölder’s and Minkowski’s Inequalities  13 Remarks 2.8. – For an x := {xk }∞ k=1 ∈ lp , ‖x‖p = 0 ⇔ x = 0. ∞ – Observe that, for any x := {xk }∞ k=1 , y := {yk }k=1 ∈ lp , ρp (x, y) = ‖x − y‖p , where ∞ ∞ x − y = {xk }∞ k=1 − {yk }k=1 := {xk − yk }k=1 . Exercise 2.7. Verify. Theorem 2.4 (Minkowski’s Inequality for Sequences). Let 1 ≤ p ≤ ∞. Then, for any ∞ ∞ x := {xk }∞ k=1 , y := {yk }k=1 ∈ lp , x + y := {xk + yk }k=1 ∈ lp and ‖x + y‖p ≤ ‖x‖p + ‖y‖p . Proof. The cases of p = 1 or p = ∞ are trivial. Exercise 2.8. Verify. For an arbitrary 1 < p < ∞ and each n ∈ ℕ, by Minkowski’s inequality for ntuples, n 1/p [ ∑ xk + yk p ] k=1 n 1/p ≤ [ ∑ xk p ] k=1 1/p n + [ ∑ yk p ] k=1 ∞ 1/p ≤ [ ∑ xk p ] k=1 ∞ 1/p + [ ∑ yk p ] k=1 . Passing to the limit as n → ∞, we infer both the convergence for the series + yk p , i. e., the fact that x + y ∈ lp , and the desired inequality. ∑∞ k=1 xk Exercise 2.9. Applying Minkowski’s inequality (Theorems 2.3 and 2.4), verify Examples 2.1, 3 and 4, for 1 < p < ∞. Theorem 2.5 (Hölder’s Inequality for Sequences). Let 1 ≤ p, q ≤ ∞ be conjugate in∞ dices. Then, for any x = {xk }∞ k=1 ∈ lp and any y = {yk }k=1 ∈ lq , the product sequence ∞ {xk yk }k=1 ∈ l1 and ∞ ∑ xk yk  ≤ ‖x‖p ‖y‖q . k=1 Exercise 2.10. Prove based on Minkowski’s Inequality for nTuples (Theorem 2.2) similarly to proving Minkowski’s Inequality for Sequences (Theorem 2.4). Remark 2.9. The important special cases of Hölder’s inequality with p = q = 2: n n k=1 k=1 ∞ ∞ k=1 k=1 1/2 1/2 n ∑ xk yk  ≤ [ ∑ xk 2 ] [ ∑ yk 2 ] 1/2 (n ∈ ℕ), k=1 ∞ 1/2 ∑ xk yk  ≤ [ ∑ xk 2 ] [ ∑ yk 2 ] k=1 (2.2) 14  2 Metric Spaces are known as the Cauchy4 –Schwarz5 inequalities (2.2) for ntuples and sequences, respectively. 2.3 Subspaces of a Metric Space Definition 2.5 (Subspace of a Metric Space). If (X, ρ) is a metric space and Y ⊆ X, then the restriction of the metric ρ(⋅, ⋅) to Y × Y is a metric on Y and the metric space (Y, ρ) is called a subspace of (X, ρ). Examples 2.3. 1. Any nonempty subset of ℝn or ℂn (n ∈ ℕ) is a metric space relative to the pmetric (1 ≤ p ≤ ∞). 2. The sets of real/complex sequences c00 := {x := {xk }∞ (eventually zero sequences), k=1 ∃N ∈ ℕ : xk = 0, k ≥ N} c0 := {x := {xk }∞ (vanishing sequences), k=1 lim xk = 0} k→∞ c := {x := {xk }∞ (convergent sequences) k=1 lim xk ∈ ℝ (or ℂ)} k→∞ endowed with the supremum metric ∞ x := {xk }∞ k=1 , y := {yk }k=1 → ρ∞ (x, y) := sup xk − yk  k∈ℕ are subspaces of the space of bounded sequences l∞ due to the settheoretic inclusions c00 ⊂ lp ⊂ lq ⊂ c0 ⊂ c ⊂ l∞ , where 1 ≤ p < q < ∞. Exercise 2.11. Verify the inclusions and show that they are proper. 2.4 Function Spaces The following gives more examples of metric spaces. Examples 2.4. 1. Let T be a nonempty set. The set M(T) of all real/complexvalued functions bounded on T, i. e., all functions f : T → ℝ (or ℂ) such that sup f (t) < ∞, t∈T 4 AugustinLouis Cauchy (1789–1857). 5 Karl Hermann Amandus Schwarz (1843–1921). 2.4 Function Spaces  15 is a metric space relative to the supremum metric (or uniform metric) M(T) ∋ f , g → ρ∞ (f , g) := sup f (t) − g(t). t∈T (n) Remark 2.10. The spaces l∞ (n ∈ ℕ) and l∞ are particular cases of M(T) with T = {1, . . . , n} and T = ℕ, respectively. 2. The set C[a, b] of all real/complexvalued functions continuous on an interval [a, b] (−∞ < a < b < ∞) is a metric space relative to the maximum metric (or uniform metric) C[a, b] ∋ f , g → ρ∞ (f , g) = max f (t) − g(t) a≤t≤b as a subspace of M[a, b]. Exercise 2.12. Answer the following questions: (a) Why can max can be used instead of sup for C[a, b]? (b) Can sup be replaced with max for the uniform metric on M[a, b]? 3. For any −∞ < a < b < ∞, the set P of all polynomials with real/complex coefficients and the set Pn of all polynomials of degree at most n (n = 0, 1, 2, . . . ) are metric spaces as subspaces of (C[a, b], ρ∞ ) due to the following (proper) inclusions: Pm ⊂ Pn ⊂ P ⊂ C[a, b] ⊂ M[a, b], where 0 ≤ m < n. 4. The set C[a, b] (−∞ < a < b < ∞) is a metric space relative to the integral metric b C[a, b] ∋ f , g → ρ1 (f , g) = ∫ f (t) − g(t) dt. a However, the extension of the latter to the wider set R[a, b] of real/complexvalued functions Riemann integrable on [a, b] is only a semimetric. Exercise 2.13. Verify both statements. 5. The set BV[a, b] (−∞ < a < b < ∞) of real/complexvalued functions of bounded variation on [a, b] (−∞ < a < b < ∞), i. e., all functions f : [a, b] → ℝ (or ℂ) such that the total variation of f over [a, b] n Vab (f ) := sup ∑ f (tk ) − f (tk−1 ) < ∞, P k=1 where the supremum is taken over all partitions P: a = t0 < t1 < ⋅ ⋅ ⋅ < tn = b of [a, b], is a metric space relative to the metric BV[a, b] ∋ f , g → ρ(f , g) := f (a) − g(a) + Vab (f − g). 16  2 Metric Spaces Exercise 2.14. (a) Verify. (b) For which functions in BV[a, b] is Vab (f ) = 0? (c) Show that d(f , g) := Vab (f − g), f , g ∈ BV[a, b], is only a semimetric on BV[a, b]. Remark 2.11. When contextually important to distinguish between the real and complex cases, we can use the notations M(T, ℝ), C([a, b], ℝ), BV([a, b], ℝ) and M(T, ℂ), C([a, b], ℂ), BV([a, b], ℂ), respectively. 2.5 Further Properties of Metric Theorem 2.6 (Generalized Triangle Inequality). In a metric space (X, ρ), for any finite collection of points {x1 , . . . , xn } ⊆ X (n ∈ ℕ, n ≥ 3), ρ(x1 , xn ) ≤ ρ(x1 , x2 ) + ρ(x2 , x3 ) + ⋅ ⋅ ⋅ + ρ(xn−1 , xn ). Exercise 2.15. Prove by induction. Theorem 2.7 (Inverse Triangle Inequality). In a metric space (X, ρ), for arbitrary x, y, z ∈ X, ρ(x, y) − ρ(y, z) ≤ ρ(x, z). Proof. Let x, y, z ∈ X be arbitrary. On one hand, by the triangle inequality and symmetry, ρ(x, y) ≤ ρ(x, z) + ρ(z, y) = ρ(x, z) + ρ(y, z), which implies ρ(x, y) − ρ(y, z) ≤ ρ(x, z). (2.3) On the other hand, in the same manner, ρ(y, z) ≤ ρ(y, x) + ρ(x, z) = ρ(x, y) + ρ(x, z), which implies ρ(y, z) − ρ(x, y) ≤ ρ(x, z). Jointly, inequalities (2.3) and (2.4) are equivalent to the desired one. (2.4) 2.6 Convergence and Continuity  17 Theorem 2.8 (Quadrilateral Inequality). In a metric space (X, ρ), for arbitrary x, y, u, v ∈ X, ρ(x, y) − ρ(u, v) ≤ ρ(x, u) + ρ(y, v). Proof. For any x, y, u, v ∈ X, ρ(x, y) − ρ(u, v) = ρ(x, y) − ρ(y, u) + ρ(y, u) − ρ(u, v) ≤ ρ(x, y) − ρ(y, u) + ρ(y, u) − ρ(u, v) by the Inverse Triangle Inequality (Theorem 2.7); ≤ ρ(x, u) + ρ(y, v). Remark 2.12. With only the symmetry axiom and the triangle inequality used in the proofs of this section’s statements, they are, obviously, true for a semimetric. 2.6 Convergence and Continuity The notion of metric brings to life the important concepts of limit and continuity. 2.6.1 Convergence of a Sequence Definition 2.6 (Limit and Convergence of a Sequence). A sequence of points {xn }∞ n=1 in a metric space (X, ρ) is said to converge (to be convergent) to a point x ∈ X if ∀ ε > 0 ∃ N ∈ ℕ ∀ n ≥ N : ρ(xn , x) < ε, i. e., lim ρ(xn , x) = 0 (ρ(xn , x) → 0, n → ∞). n→∞ We write in this case lim x n→∞ n = x or xn → x, n → ∞ and say that x is the limit of {xn }∞ n=1 . A sequence {xn }∞ in a metric space (X, ρ) is called convergent if it converges to n=1 some x ∈ X and divergent otherwise. Theorem 2.9 (Uniqueness of a Limit). The limit of a convergent sequence {xn }∞ n=1 in a metric space (X, ρ) is unique. Exercise 2.16. Prove. 18  2 Metric Spaces Examples 2.5 (Convergence in Concrete Metric Spaces). 1. A sequence is convergent in a discrete space (X, ρd ) iff it is eventually constant. 2. Convergence of a sequence in the space lp(n) (n ∈ ℕ and 1 ≤ p ≤ ∞) is equivalent to componentwise convergence, i. e., (x1(k) , . . . , xn(k) ) → (x1 , . . . , xn ), k → ∞ ⇔ ∀ i = 1, . . . , n : xi(k) → xi , k → ∞. 3. Convergence of a sequence in the space lp (1 ≤ p ≤ ∞) ∞ {xk(n) }k=1 =: x(n) → x := {xk }∞ k=1 , n → ∞, implies termwise convergence, i. e., ∀ k ∈ ℕ : xk(n) → xk , n → ∞. 4. Convergence in (M(T), ρ∞ ) (see Examples 2.4) is the uniform convergence on T, i. e., fn → f , n → ∞, in (M(T), ρ∞ ) iff ∀ ε > 0 ∃ N ∈ ℕ ∀ n ≥ N ∀ t ∈ T : fn (t) − f (t) < ε. The same is true for l∞ as a particular case of M(T) with T = ℕ and for (C[a, b], ρ∞ ) as a subspace of (M[a, b], ρ∞ ). Exercise 2.17. (a) Verify each statement and give corresponding examples. (b) Give an example showing that the converse to 3 is not true, i. e., termwise convergent is necessary, but not sufficient for convergence in lp (1 ≤ p ≤ ∞) (cf. Section 2.19, Problems 6 and 7). 2.6.2 Continuity, Uniform Continuity, and Lipschitz Continuity Here, we introduce three notions of continuity in the order of increasing strength: regular, uniform, and Lipschitz. Definition 2.7 (Continuity of a Function). Let (X, ρ) and (Y, σ) be metric spaces. A function f : X → Y is called continuous at a point x0 ∈ X if ∀ ε > 0 ∃ δ > 0 ∀ x ∈ X with ρ(x, x0 ) < δ : σ(f (x), f (x0 )) < ε. A function f : X → Y is called continuous on X if it is continuous at every point of X. The set of all such functions is designated as C(X, Y) and we write f ∈ C(X, Y). 2.6 Convergence and Continuity  19 Remarks 2.13. – When X and Y are subsets of ℝ with the regular distance, we obtain the familiar calculus (ε, δ)definitions. – When Y = ℝ or Y = ℂ, the shorter notation C(X) is often used. It is convenient to characterize continuity in terms of sequences. Theorem 2.10 (Sequential Characterization of Local Continuity). Let (X, ρ) and (Y, σ) be metric spaces. A function f : X → Y is continuous at a point x0 ∈ X iff, for each sequence {xn }∞ n=1 in X such that lim x n→∞ n = x0 in (X, ρ), we have: lim f (xn ) = f (x0 ) in (Y, σ). n→∞ Exercise 2.18. Prove. Hint. The necessity is proved directly. Prove the sufficiency by contrapositive. Using the Sequential Characterization of Local Continuity (Theorem 2.10), one can easily prove the following two theorems. Theorem 2.11 (Properties of Numeric Continuous Functions). Let (X, ρ) be a metric space and Y = ℝ or Y = ℂ with the regular distance. If f and g are continuous at a point x0 ∈ X, then (1) ∀ c ∈ ℝ (or c ∈ ℂ), cf is continuous at x0 , (2) f + g is continuous at x0 , (3) f ⋅ g is continuous at x0 , (4) Provided g(x0 ) ≠ 0, gf is continuous at x0 . Theorem 2.12 (Continuity of Composition). Let (X, ρ), (Y, σ), and (Z, τ), f : X → Y and g : Y → Z. If for some x0 ∈ X f is continuous at x0 and g is continuous at y0 = f (x0 ), then the composition g(f (x)) is continuous at x0 . Exercise 2.19. Prove Theorems 2.11 and 2.12 using the sequential approach. Remark 2.14. The statements of Theorems 2.11 and 2.12 are naturally carried over to functions continuous on the whole space (X, ρ). Definition 2.8 (Uniform Continuity). Let (X, ρ) and (Y, σ) be metric spaces. A function f : X → Y is said to be uniformly continuous on X if ∀ ε > 0 ∃ δ > 0 ∀ x , x ∈ X with ρ(x , x ) < δ : σ(f (x ), f (x )) < ε. 20  2 Metric Spaces Remark 2.15. As the following example shows, a uniformly continuous function is continuous, but not vice versa. Example 2.6. For X = Y = (0, ∞), f (x) = x is uniformly continuous and f (x) = continuous, but not uniformly. 1 x is Exercise 2.20. Verify. Definition 2.9 (Lipschitz Continuity). Let (X, ρ) and (Y, σ) be metric spaces. A function f : X → Y is said to be Lipschitz continuous on X with Lipschitz constant L if6 ∃ L ≥ 0 ∀ x , x ∈ X : σ(f (x ), f (x )) ≤ Lρ(x , x ). Remarks 2.16. – The smallest Lipschitz constant is called the best Lipschitz constant. – A constant function is Lipschitz continuous with the best Lipschitz constant L = 0. – By the Mean Value Theorem, a realvalued differentiable function f on an interval I ⊆ ℝ is Lipschitz continuous on I iff its derivative f is bounded on I. In particular, all functions in C 1 [a, b] (−∞ < a < b < ∞) are Lipschitz continuous on [a, b]. – As the following example shows, a Lipschitz continuous function is uniformly continuous, but not vice versa. Example 2.7. For X = Y = [0, 1] with the regular distance, the function f (x) := √x is uniformly continuous on [0, 1], as follows from the Heine–Cantor Uniform Continuity Theorem (Theorem 2.44), but is not Lipschitz continuous on [0, 1]. Exercise 2.21. Verify. 2.7 Balls, Separation, and Boundedness The geometric concepts of balls and spheres, generalizing their familiar counterparts, are rather handy as well as is a generalized notion of boundedness. Definition 2.10 (Balls and Spheres). Let (X, ρ) be a metric space and r ≥ 0. – The open ball of radius r centered at a point x0 ∈ X is the set B(x0 , r) := {x ∈ X  ρ(x, x0 ) < r} . – The closed ball of radius r centered at a point x0 ∈ X is the set B(x0 , r) := {x ∈ X  ρ(x, x0 ) ≤ r} . 6 Sigismund Lipschitz (1832–1903). 2.7 Balls, Separation, and Boundedness  21 – The sphere of radius r centered at a point x0 ∈ X is the set S(x0 , r) := {x ∈ X  ρ(x, x0 ) = r} = B(x0 , r) \ B(x0 , r). Remarks 2.17. – When contextually important to indicate which space the balls/spheres are considered in, the letter designating the space in question is added as a subscript. E. g., for (X, ρ), we use the notations BX (x0 , r), – BX (x0 , r), and SX (x0 , r), x0 ∈ X, r ≥ 0. As is easily seen, for an arbitrary x0 ∈ X, B(x0 , 0) = 0 and B(x0 , 0) = S(x0 , 0) = {x0 } (trivial cases). Exercise 2.22. (a) Explain the latter. (b) Describe balls and spheres in ℝ and ℂ with the regular distance, give some examples. (c) Sketch the unit sphere S(0, 1) in (ℝ2 , ρ1 ), (ℝ2 , ρ2 ), and (ℝ2 , ρ∞ ). (d) Describe balls and spheres in (C[a, b], ρ∞ ). (e) Let (X, ρd ) be a discrete metric space and x0 ∈ X be arbitrary. Describe B(x0 , r), B(x0 , r), and S(x0 , r) for different values of r ≥ 0. Proposition 2.1 (Separation Property). Let (X, ρ) be metric space. Then ∀ x, y ∈ X, x ≠ y ∃ r > 0 : B(x, r) ∩ B(y, r) = 0. i. e., distinct points in a metric space can be separated by disjoint balls. Exercise 2.23. Prove. The definitions of convergence and continuity can be naturally reformulated in terms of balls. Definition 2.11 (Equivalent Definitions of Convergence and Continuity). A sequence of points {xn }∞ n=1 in a metric space (X, ρ) is said to converge (to be convergent) to a point x ∈ X if ∀ ε > 0 ∃ N ∈ ℕ ∀ n ≥ N : xn ∈ B(x0 , ε), in which case we say that the sequence {xn }∞ n=1 is eventually in the εball B(x, ε). Let (X, ρ) and (Y, σ) be metric spaces. A function f : X → Y is called continuous at a point x0 ∈ X if ∀ ε > 0 ∃ δ > 0 : f (BX (x0 , δ)) ⊆ BY (f (x0 ), ε). 22  2 Metric Spaces Definition 2.12 (Bounded Set). Let (X, ρ) be a metric space. A nonempty set A ⊆ X is called bounded if diam(A) := sup ρ(x, y) < ∞. x,y∈A The number diam(A) is called the diameter of A. Remark 2.18. The empty set 0 is regarded to be bounded with diam(0) := 0. Examples 2.8. 1. In a metric space (X, ρ), an open/closed ball of radius r > 0 is a bounded set of diameter at most 2r. 2. In (ℝ, ρ), the sets (0, 1], {1/n}n∈ℕ are bounded and the sets (−∞, 1), {n2 }n∈ℕ are not. 3. In l∞ , the set {(xn )n∈ℕ  xn  ≤ 1, n ∈ ℕ} is bounded and, in lp (1 < p < ∞), it is not. 4. In (C[0, 1], ρ∞ ), the set {t n }n∈ℤ+ is bounded and, in (C[0, 2], ρ∞ ), it is not. Exercise 2.24. (a) Verify. (b) Show that a set A is bounded iff it is contained in some (closed) ball, i. e., ∃ x ∈ X ∃ r ≥ 0 : A ⊆ B(x, r). (c) Describe all bounded sets in a discrete metric space (X, ρd ). (d) Give an example of a metric space (X, ρ), in which, for a ball B(x, r) with some x ∈ X and r > 0, diam(B(x, r)) < 2r. Theorem 2.13 (Properties of Bounded Sets). The bounded sets in a metric space (X, ρ) have the following properties: (1) a subset of a bounded set is bounded; (2) an arbitrary intersection of bounded sets is bounded; (3) a finite union of bounded sets is bounded. Exercise 2.25. (a) Prove. (b) Give an example showing that an infinite union of bounded sets need not be bounded. Definition 2.13 (Bounded Function). Let T be a nonempty set and (X, ρ) be a metric space. A function f : T → X is called bounded if the set of its values f (T) is bounded in (X, ρ). Remark 2.19. As a particular with for T = ℕ, we obtain the definition of a bounded sequence. Brought to you by  Cambridge University Library Authenticated Download Date  10/11/18 9:53 PM 2.8 Interior Points, Open Sets  23 2.8 Interior Points, Open Sets Now, we are ready to define openness and closedness for sets. Definition 2.14 (Interior Point). Let (X, ρ) be a metric space. A point x ∈ X is called an interior point of a nonempty set A ⊆ X if A contains a nontrivial open ball centered at x, i. e., ∃ r > 0 : B(x, r) ⊆ A. Examples 2.9. 1. In an arbitrary metric space (X, ρ), any point x ∈ X is, obviously, an interior point of an open ball B(x, r) or a closed ball B(x, r) with an arbitrary r > 0. 2. For the set [0, 1) in ℝ with the regular distance, the points 0 < x < 1 are interior and the point x = 0 is not. 3. A singleton {x} in ℝ with the regular distance, has no interior points. Exercise 2.26. Verify. Definition 2.15 (Interior of a Set). The interior of a nonempty set A in a metric space (X, ρ) is the set of all interior points of A. Notation. int(A). Remark 2.20. Thus, we have always the inclusion int(A) ⊆ A, the prior examples showing that the inclusion may be proper and that int(A) may be empty. Exercise 2.27. Give some examples. Definition 2.16 (Open Set). A nonempty set A in a metric space (X, ρ) is called open if each point of A is its interior point, i. e., A = int(A). Remark 2.21. The empty set 0 is regarded to be open and the whole space X is trivially open as well. Exercise 2.28. (a) Verify that, in ℝ with the regular distance, the intervals of the form (a, ∞), (−∞, b), and (a, b) (−∞ < a < b < ∞) are open sets. (b) Prove that, in a metric space (X, ρ), an open ball B(x0 , r) (x0 ∈ X, r ≥ 0) is an open set. (c) Describe all open sets in a discrete metric space (X, ρd ). The concept of openness in a metric space can be characterized sequentially. Brought to you by  Cambridge University Library Authenticated Download Date  10/11/18 9:53 PM 24  2 Metric Spaces Theorem 2.14 (Sequential Characterizations of Open Sets). Let (X, ρ) be a metric space. A set A ⊆ X is open in (X, ρ) iff, for any sequence {xn }∞ n=1 ⊆ X convergent to a point x0 ∈ A, {xn }∞ is eventually in A. n=1 Exercise 2.29. Prove. Theorem 2.15 (Properties of Open Sets). The open sets in a metric space (X, ρ) have the following properties: (1) 0 and X are open sets; (2) an arbitrary union of open sets is open; (3) an arbitrary finite intersection of open sets is open. Exercise 2.30. (a) Prove. (b) Give an example showing that an infinite intersection of open sets need not be open. Definition 2.17 (Metric Topology). The collection G of all open sets in a metric space (X, ρ) is called the metric topology generated by the metric ρ. 2.9 Limit Points, Closed Sets Definition 2.18 (Limit Point/Derived Set). Let (X, ρ) be a metric space. A point x ∈ X is called a limit point (also an accumulation point or a cluster point) of a set A in X if every open ball centered at x contains a point of A distinct from x, i. e., ∀ r > 0 : B(x, r) ∩ (A \ {x}) ≠ 0. The set A of all limit points of A is called the derived set of A. Remarks 2.22. – A limit point x of a set A need not belong to A. It may even happen that none of them does, i. e., A ⊆ Ac . – – Each open ball centered at a limit point x of a set A in a metric space (X, ρ) contains infinitely many points of A distinct from x0 . To have a limit point, a set A in a metric space (X, ρ) must necessarily be nonempty and even infinite. However, an infinite set need not have limit points. Exercise 2.31. (a) Verify and give corresponding examples. (b) Describe the situation in a discrete metric space (X, ρd ). Brought to you by  Cambridge University Library Authenticated Download Date  10/11/18 9:53 PM 2.9 Limit Points, Closed Sets  25 (c) Give examples showing that an interior point of a set need not be its limit point and vise versa. Limit points can be characterized sequentially as follows. Theorem 2.16 (Sequential Characterization of Limit Points). Let (X, ρ) be a metric space. A point x ∈ X is a limit point of a set A ⊆ X iff A contains a sequence of points {xn }∞ n=1 distinct from x convergent to x, i. e., x ∈ A ⇔ ∃ {xn }∞ n=1 ⊆ A \ {x} : xn → x, n → ∞. Exercise 2.32. Prove. Definition 2.19 (Isolated Point). Let (X, ρ) be a metric space. A point x of a set A in X, which is not its limit point, is called an isolated point of A, i. e., there is an open ball centered at x containing no other points of A, but x. Exercise 2.33. Show that a function f from a metric space (X, ρ) to a metric space (Y, σ) is continuous at isolated points of X, if any. Definition 2.20 (Closure of a Set). The closure A of a set A in a metric space (X, ρ) is the set consisting of all points, which are either points of A or limit points of A, i. e., A := A ∪ A . Remarks 2.23. – Obviously 0 = 0, and hence, 0 = 0. – We always have the inclusion A ⊆ A, – which may be proper. A point x ∈ A iff every nontrivial open ball centered at x contains a point of A (not necessarily distinct from x). Exercise 2.34. Verify and give a corresponding example. By the definition and the Sequential Characterization of Limit Points (Theorem 2.16), we obtain the following Theorem 2.17 (Sequential Characterization of a Closure). Let (X, ρ) be a metric space. For a set A ⊆ X, x ∈ A ⇔ ∃ {xn }∞ n=1 ⊆ A : xn → x, n → ∞. Brought to you by  Cambridge University Library Authenticated Download Date  10/11/18 9:53 PM 26  2 Metric Spaces Exercise 2.35. (a) Prove. (b) Is it true that, in any metric space (X, ρ), for each x ∈ X and r > 0, B(x, r) = B(x, r)? (c) Let A be a set in a metric space (X, ρ). Prove that, for each open set O in (X, ρ), O ∩ A ≠ 0 ⇔ O ∩ A ≠ 0. Definition 2.21 (Closed Set). Let (X, ρ) be a metric space. A set A in X is called closed if it contains all its limit points, i. e., A ⊆ A, and hence, A = A. Remarks 2.24. – The whole space X is trivially closed. – Also closed are the sets with no limit points, in particular, finite sets, including the empty set 0. – A set in a metric space (X, ρ), which is simultaneously closed and open is called clopen. There are always at least two (trivial) clopen sets: 0 and X. However, there can exist nontrivial ones. Exercise 2.36. (a) Verify that in ℝ with the regular distance the intervals of the form [a, ∞), (−∞, b], and [a, b] (−∞ < a < b < ∞) are closed sets. (b) Verify that the sets (0, 1) and {2} are clopen in the metric space (0, 1) ∪ {2} with the regular distance. (c) Describe all closed sets in a discrete metric space (X, ρd ). Theorem 2.18 (Characterizations of Closed Sets). Let (X, ρ) be a metric space and A ⊆ X. Then the following statements are equivalent: 1. A is closed in (X, ρ). 2. The complement Ac is open. 3. (Sequential Characterization) For any sequence {xn }∞ n=1 ⊆ A convergent in (X, ρ), limn→∞ xn ∈ A, i. e., A contains the limits of all its convergent sequences. Exercise 2.37. (a) Prove. (b) Show in two different ways that, in a metric space (X, ρ), a closed ball B(x, r) (x ∈ X, r ≥ 0) is a closed set. The properties of the closed sets follow immediately from the properties of the open sets via de Morgan’s laws (see Section 1.1.1) considering the fact that the closed and open sets are complementary. Brought to you by  Cambridge University Library Authenticated Download Date  10/11/18 9:53 PM 2.10 Dense Sets and Separable Spaces  27 Theorem 2.19 (Properties of Closed Sets). The closed sets in a metric space (X, ρ) have the following properties: (1) 0 and X are closed sets; (2) an arbitrary intersection of closed sets is closed; (3) a finite union of closed sets is closed. Exercise 2.38. (a) Prove. (b) Give an example showing that an infinite union of closed sets need not be closed. 2.10 Dense Sets and Separable Spaces Here, we consider the notions of the denseness of a set in a metric space, i. e., of a set’s points being able to be found arbitrarily close to all points of the space, and of the separability of a metric space, i. e., of a space’s containing a countable such a set. Definition 2.22 (Dense Set). A set A in a metric space (X, ρ) is called dense if A = X. Remark 2.25. Thus, a set A is dense in a metric space (X, ρ) iff an arbitrary nontrivial open ball contains a point of A (see Remarks 2.23). Example 2.10. The set ℚ of the rational numbers is dense in ℝ. Exercise 2.39. Verify. From the Sequential Characterization of Closure (Theorem 2.17), immediately follows Theorem 2.20 (Sequential Characterization of Dense Sets). A set A is dense in a metric space (X, ρ) iff ∀x ∈ X ∃ {xn }∞ n=1 ⊆ A : xn → x, n → ∞. Definition 2.23 (Separable Metric Space). A metric space (X, ρ) containing a countable dense subset is a called separable. Remark 2.26. Any countable metric space is, obviously, separable. However, as the following examples show, a metric space need not be countable to be separable. Examples 2.11. 1. The spaces lp(n) are separable for (n ∈ ℕ, 1 ≤ p ≤ ∞), which includes the cases of ℝ and ℂ with the regular distances. Indeed, as a countable dense set here, one can consider that of all ordered ntuples with (real/complex) rational components. 2. The spaces lp are separable for 1 ≤ p < ∞. Brought to you by  Cambridge University Library Authenticated Download Date  10/11/18 9:53 PM 28  2 Metric Spaces Indeed, as a countable dense set here, one can consider that of all eventually zero sequences with (real/complex) rational terms. 3. The space (C[a, b], ρ∞ ) is separable, which follows from Weierstrass Approximation Theorem (Theorem 2.49) when we consider as a countable dense set that of all polynomials with rational coefficients. 4. More examples of separable spaces can be built based on the fact that any subspace of a separable metric space is separable (Proposition 2.19) (see Section 2.19, Problem 17). 5. The space l∞ is not separable. Exercise 2.40. (a) Verify 1–3 using the Properties of Countable Sets (Theorem 1.3). (b) Prove 5. Hint. In l∞ , consider the uncountable set B of all binary sequences, i. e., the sequences whose only entries are 0 or 1 (see the Uncountable Sets Proposition (Proposition 1.1)). The balls of the uncountable collection {B(x, 1/2)  x ∈ B} are pairwise disjoint. Explain the latter and, assuming that there exists a countable dense subset in l∞ , arrive at a contradiction. (c) Prove that the spaces (C[a, b], ρp ) (1 ≤ p < ∞) (see Section 2.19, Problem 4) are separable. (d) In which case is a discrete metric space (X, ρd ) separable? More facts on separability are stated as problems in Section 2.19. 2.11 Exterior and Boundary Relative to a set A in a metric space (X, ρ), each point of the space falls into one of the tree pairwise disjoint classes: interior, exterior, or boundary. The interior points having been defined above (see Definition 2.14), it remains to define the exterior and boundary ones. Definition 2.24 (Exterior and Boundary Points). Let A be a set in a metric space (X, ρ). – We say that x ∈ X is an exterior point of A if it is an interior point of the complement Ac := X \ A, i. e., there is an open ball centered at x0 contained in Ac . All exterior points of a set A form its exterior ext(A). – We say that x ∈ X is a boundary point of A if it is neither interior nor exterior point of A, i. e., every open ball centered at x contains both a point of A and a point of Ac . All boundary points of a set A form its boundary 𝜕A. Brought to you by  Cambridge University Library Authenticated Download Date  10/11/18 9:53 PM 2.12 Equivalent Metrics, Homeomorphisms and Isometries  29 Remark 2.27. By definition, for each set A in a metric space (X, ρ), int(A), ext(A), and 𝜕A form a partition of X, i. e., are pairwise disjoint and int(A) ∪ 𝜕A ∪ ext(A) = X. Exercise 2.41. (a) In an arbitrary metric space (X, ρ), determine int(0), ext(0), 𝜕 0 and int(X), ext(X), 𝜕X. (b) Determine int(A), ext(A), and 𝜕A of a nonempty proper subset A of a discrete metric space (X, ρd ). (c) Determine int(ℚ), ext(ℚ), and 𝜕 ℚ in ℝ. 2.12 Equivalent Metrics, Homeomorphisms and Isometries 2.12.1 Equivalent Metrics Definition 2.25 (Equivalent Metrics). Two metrics ρ1 and ρ2 on a nonempty set X are called equivalent if they generate the same metric topology. Exercise 2.42. Show that two metrics ρ1 and ρ2 on a nonempty set X are equivalent iff (a) for an arbitrary x ∈ X, each open ball centered at x relative to ρ1 contains an open ball centered at x relative to ρ2 and vice versa; (b) any sequence {xn }∞ n=1 convergent relative to ρ1 converges to the same limit relative to ρ2 and vice versa. Exercise 2.43. (a) Show that the equivalence of metrics on a nonempty set X is an equivalence relation (reflexive, symmetric, and transitive) on the set of all metrics on X. (b) Show that, for any metric space (X, ρ), ρ is equivalent to the standard bounded metric d(x, y) := min(ρ(x, y), 1), x, y ∈ X (see Section 2.19, Problem 3). (c) Show that, if, for two metrics ρ1 and ρ2 on a nonempty set X, ∃ c, C > 0 : cρ1 (x, y) ≤ ρ2 (x, y) ≤ Cρ1 (x, y), x, y ∈ X, (2.5) ρ1 and ρ2 are equivalent. Use the prior example to show that the converse is not true. (d) Show that, on the nspace ℝn or ℂn (n ∈ ℕ), all pmetrics (1 ≤ p ≤ ∞) (see Examples 2.1) are equivalent in the sense of (2.5). Hint. Show the equivalence of any pmetric ρp (1 ≤ p < ∞) in the sense of (2.5) to ρ∞ . Brought to you by  Cambridge University Library Authenticated Download Date  10/11/18 9:53 PM 30  2 Metric Spaces 2.12.2 Homeomorphisms and Isometries Definition 2.26 (Homeomorphism of Metric Spaces). A homeomorphism of a metric space (X, ρ) to a metric space (Y, σ) is a mapping T : X → Y, which is bijective and bicontinuous, i. e., T ∈ C(X, Y) and the inverse T −1 ∈ C(Y, X). The space (X, ρ) is said to be homeomorphic to (Y, σ). Remarks 2.28. – The relation of being homeomorphic to is an equivalence relation on the set of all metric spaces, and thus, we can say that homeomorphism is between the spaces. – Homeomorphic metric spaces are topologically indistinguishable, i. e., have the same topological properties such as separability or the existence of nontrivial clopen sets (disconnectedness (see, e. g., [38, 41])). – Two metrics ρ1 and ρ2 on a nonempty set X are equivalent iff the identity mapping Ix := x, x ∈ X, is a homeomorphism between the spaces (X, ρ1 ) and (X, ρ2 ). Exercise 2.44. Verify. To show that two metric spaces are homeomorphic, one needs to specify a homeomorphism between them; whereas to show that they are not homeomorphic, one only needs to specify a topological property not shared by them. Examples 2.12. 1. Any open bounded interval (a, b) (−∞ < a < b < ∞) is homeomorphic to (0, 1) via f (x) := 2. x−a . b−a An open interval of the form (−∞, b) in ℝ is homeomorphic to (−b, ∞) via f (x) := −x. 3. An open interval of the form (a, ∞) in ℝ is homeomorphic to (1, ∞) via f (x) := x − a + 1. 4. The interval (1, ∞) is homeomorphic to (0, 1) via f (x) := 1 . x Brought to you by  Cambridge University Library Authenticated Download Date  10/11/18 9:53 PM 2.13 Completeness and Completion 5.  31 The interval (0, 1) is homeomorphic to ℝ via f (x) := cot(πx). 6. 7. The interval (0, 1) is not homeomorphic to the set (0, 1) ∪ {2} with the regular distance, the latter having nontrivial clopen sets (see Examples 2.36). The spaces l1 and l∞ are not homeomorphic, the former being separable and the latter being nonseparable. Remark 2.29. Thus, all open intervals in ℝ (bounded or not) are homeomorphic. The following is a very important case of a homeomorphism. Definition 2.27 (Isometry of Metric Spaces). Let (X, ρ) and (Y, σ) be metric spaces. An isometry of X to Y is a onetoone (i. e., injective) mapping T : X → Y, which is distance preserving, i. e., σ(Tx, Ty) = ρ(x, y), x, y ∈ X. It is said to isometrically embed X in Y. If an isometry T : X → Y is onto (i. e., surjective), it is called an isometry between X and Y and the spaces are called isometric. Remarks 2.30. – The relation of being isometric to is an equivalence relation on the set of all metric spaces, and thus, we can say that isometry is between the spaces. – An isometry between metric spaces (X, ρ) and (Y, σ) is, obviously, a homeomorphism between them but not vice versa (see Examples 2.12). – Isometric metric spaces are metrically indistinguishable. Exercise 2.45. Identify isometries in Examples 2.12. 2.13 Completeness and Completion Let us now deal with the concept of completeness, which is a fundamental property of metric spaces underlying many important facts. 2.13.1 Cauchy/Fundamental Sequences Definition 2.28 (Cauchy/Fundamental Sequence). A sequence {xn }∞ n=1 in a metric space (X, ρ) is called a Cauchy sequence, or a fundamental sequence, if ∀ ε > 0 ∃ N ∈ ℕ ∀ m, n ≥ N : ρ(xm , xn ) < ε. Brought to you by  Cambridge University Library Authenticated Download Date  10/11/18 9:53 PM 32  2 Metric Spaces Remark 2.31. The latter is equivalent to ρ(xm , xn ) → 0, m, n → ∞ or to sup ρ(xn+k , xn ) → 0, n → ∞. k∈ℕ Examples 2.13. 1. A sequence is fundamental in a discrete space (X, ρd ) iff it is eventually constant. ∞ 2. The sequence {1/n}∞ n=1 is fundamental in ℝ and the sequence {n}n=1 is not. ∞ 3. The sequence {xn := {1, 1/2, . . . , 1/n, 0, 0, . . . }}n=1 is fundamental in (c00 , ρ∞ ) and in lp (1 < p < ∞), but not in l1 . ∞ 4. The sequence {en := {δnk }∞ k=1 }n=1 , where δnk is the Kronecker delta, is fundamental neither in (c00 , ρ∞ ) nor in lp (1 ≤ p ≤ ∞). Exercise 2.46. Verify. Theorem 2.21 (Properties of Fundamental Sequences). In a metric space (X, ρ), (1) every fundamental sequence {xn }∞ n=1 is bounded, (2) every convergent sequence {xn }∞ n=1 is fundamental, ∞ (3) if a sequence {xn }n=1 is fundamental, then any sequence {yn }∞ n=1 asymptotically equivalent to {xn }∞ in the sense that n=1 ρ(xn , yn ) → 0, n → ∞, is also fundamental. Exercise 2.47. Prove. Remark 2.32. A fundamental sequence need not converge. Thus, the sequence {xn = {1, 1/2, . . . , 1/n, 0, 0, . . . }}∞ n=1 is fundamental, but divergent in (c00 , ρ∞ ). It does converge to x = {1/n}∞ n=1 , however, in the wider space (c0 , ρ∞ ). Exercise 2.48. Verify. Proposition 2.2 (Fundamentality and Uniform Continuity). Let (X, ρ) and (Y, σ) be metric spaces and a function f : X → Y be uniformly continuous on X. If {xn }∞ n=1 is a fundamental sequence in (X, ρ), then {f (xn )}∞ is a fundamental sequence in (Y, σ), n=1 i. e., a uniformly continuous function maps a fundamental sequence to a fundamental sequence. Exercise 2.49. (a) Prove. (b) Give an example showing that a continuous function need not preserve fundamentality. Brought to you by  Cambridge University Library Authenticated Download Date  10/11/18 9:53 PM 2.13 Completeness and Completion  33 2.13.2 Complete Metric Spaces Definition 2.29 (Complete Metric Space). A metric space (X, ρ), in which every Cauchy/fundamental sequence converges, is called complete and incomplete otherwise. Examples 2.14. 1. The spaces ℝ and ℂ are complete relative to the regular distance as is known from analysis courses. 2. The spaces ℝ \ {0}, (0, 1), and ℚ are incomplete as subspaces of ℝ. 3. A discrete metric space (X, ρd ) is complete. Exercise 2.50. Verify 2 and 3. 4. Theorem 2.22 (Completeness of the nSpace). The (real or complex) space lp(n) (n ∈ ℕ, 1 ≤ p ≤ ∞) is complete. Exercise 2.51. Prove. Hint. Considering the equivalence of all pmetrics on the nspace in the sense of (2.5) (see Exercise 2.43) and Exercise 2.54, it suffices to show the completeness of the nspace relative to ρ∞ . 5. Theorem 2.23 (Completeness of lp (1 ≤ p < ∞)). The (real or complex) space lp (1 ≤ p < ∞) is complete. Proof. Let 1 ≤ p < ∞ and ∞ x(n) := {xk(n) }k=1 , n ∈ ℕ. be an arbitrary fundamental sequence in lp . Since ∀ ε > 0 ∃ N ∈ ℕ : ρp (xm , xn ) < ε, m, n ≥ N, for each k ∈ ℕ, 1/p ∞ (m) p xk − xk(n) ≤ [∑ xi(m) − xi(n) ] i=1 = ρp (xm , xn ) < ε, m, n ≥ N, which implies that, for every k ∈ ℕ, the numeric sequence {xk(n) }∞ n=1 of the kth terms is fundamental, and hence, converges, i. e., ∀ k ∈ ℕ ∃ xk ∈ ℝ (or ℂ) : xk(n) → xk , n → ∞. Brought to you by  Cambridge University Library Authenticated Download Date  10/11/18 9:53 PM 34  2 Metric Spaces Let us show that x := {xk }∞ k=1 ∈ lp . For any K ∈ ℕ and arbitrary m, n ≥ N, K p p ∞ ∑ xk(m) − xk(n) ≤ ∑ xk(m) − xk(n) = ρp (xm , xn )p < εp . k=1 k=1 Whence, fixing arbitrary K ∈ ℕ and n ≥ N and passing to the limit as m → ∞, we obtain K p ∑ xk − xk(n) ≤ εp , K ∈ ℕ, n ≥ N. k=1 Now, for an arbitrary n ≥ N, passing to the limit as K → ∞, we arrive at ∞ p ∑ xk − xk(n) ≤ εp , n ≥ N, (2.6) k=1 which, in particular, implies that x − xN ∈ lp , and hence, by Minkowski’s Inequality for Sequences (Theorem 2.4), x = (x − xN ) + xN ∈ lp . Whence, in view (2.6), we infer that ρp (x, xn ) ≤ ε, n ≥ N, which implies that xn → x, n → ∞, in lp and completes the proof. 6. Theorem 2.24 (Completeness of (M(T), ρ∞ )). The (M(T), ρ∞ ) is complete. (real or complex) space Proof. Let {fn }∞ n=1 be an arbitrary fundamental sequence in (M(T), ρ∞ ). Since ∀ ε > 0 ∃ N ∈ ℕ : ρ∞ (fm , fn ) < ε, m, n ≥ N, for each t ∈ T, fm (t) − fn (t) ≤ sup fm (s) − fn (s) = ρ∞ (fm , fn ) < ε, m, n ≥ N, s∈T which implies that, for every t ∈ T, the numeric sequence {fn (t)}∞ n=1 of the values at t is fundamental, and hence, converges, i. e., ∀ t ∈ T ∃ f (t) ∈ ℝ (or ℂ) : fn (t) → f (t), n → ∞. Brought to you by  Cambridge University Library Authenticated Download Date  10/11/18 9:53 PM 2.13 Completeness and Completion  35 Let us show that function T ∋ t → f (t) belongs to M(T). Indeed, for any t ∈ T, fm (t) − fn (t) < ε, m, n ≥ N. Whence, fixing arbitrary t ∈ T and n ≥ N and passing to the limit as m → ∞, we have: f (t) − fn (t) ≤ ε, t ∈ T, n ≥ N, i. e., sup f (t) − fn (t) ≤ ε, n ≥ N. t∈T (2.7) Therefore, sup f (t) ≤ sup f (t) − fN (t) + sup fN (t) ≤ sup fN (t) + ε < ∞, t∈T t∈T t∈T t∈T which implies that f ∈ M(T). Whence, in view (2.7), we infer that ρ∞ (f , fn ) ≤ ε, n ≥ N, which implies that fn → f , n → ∞, in (M(T), ρ∞ ) and completes the proof. As a particular case with T = ℕ, we obtain the following Corollary 2.1 (Completeness of l∞ ). The (real or complex) space l∞ is complete. 7. Theorem 2.25 (Completeness of (C[a, b], ρ∞ ) (−∞ < a < b < ∞)). The (real or complex) space (C[a, b], ρ∞ ) (−∞ < a < b < ∞) is complete. Exercise 2.52. Prove (cf. [16, Section 1.7, Exercise (i)]) 8. The space P of all polynomials with real/complex coefficients is incomplete as a subspace of (C[a, b], ρ∞ ). Exercise 2.53. Give a corresponding counterexample. 9. Proposition 2.3 (Incompleteness of (C[a, b], ρp ) (1 ≤ p < ∞)). The space (C[a, b], ρp ) (1 ≤ p < ∞) is incomplete. Brought to you by  Cambridge University Library Authenticated Download Date  10/11/18 9:53 PM 36  2 Metric Spaces Proof. let 1 ≤ p < ∞. For a fixed c ∈ (a, b) (say, c := (a + b)/2), consider the sequence {fn } in (C[a, b], ρp ) defined for all n ∈ ℕ sufficiently large so that a < c − 1/n as follows: 0 { { { fn (t) := {[n (t − c + 1/n)]1/p { { {1 for a ≤ t ≤ c − 1/n, for c − 1/n < t < c, for c ≤ t ≤ b. The sequence {fn } is fundamental in (C[a, b], ρp ). Indeed, for all sufficiently large m, n ∈ ℕ with m ≤ n, 1/p c p ρp (fm , fn ) = [ ∫ fm (t) − fn (t) dt ] [c−1/m ] c 1/p p ≤ [ ∫ fm (t) dt ] [c−1/m ] c by Minkowski’s inequality; 1/p c p + [ ∫ fn (t) dt ] [c−1/m ] 1/p = [ ∫ m (t − c + 1/m) dt ] [c−1/m ] 1/p c + [ ∫ n (t − c + 1/n) dt ] [c−1/n ] = 1 1 + . mp np Since, for any f ∈ C[a, b] and all sufficiently large n ∈ ℕ, c−1/n c b ρp (fn , f ) = [ ∫ f (t)p dt + ∫ fn (t) − f (t)p dt + ∫ 1 − f (t)p dt ] c [ a ] c−1/n 1/p , this jointly with the assumption that fn → f , n → ∞, in (C[a, b], ρp ) would imply that c b a c ∫ f (t)p dt + ∫ 1 − f (t)p dt = 0. Whence, by the continuity of f , we infer that f (t) = 0, t ∈ [a, c) and f (t) = 1, t ∈ (c, b], which contradicts the continuity of f at x = c. The obtained contradiction implies that sequence {fn } cannot converge in (C[a, b], ρp ). Hence, the metric space (C[a, b], ρp ) is incomplete. Brought to you by  Cambridge University Library Authenticated Download Date  10/11/18 9:53 PM 2.13 Completeness and Completion  37 Remark 2.33. The property of completeness is isometrically invariant (see Section 2.19, Problem 27), but is not homeomorphically invariant. For instance, the complete space ℝ is homeomorphic to the incomplete space (0, 1) (see Examples 2.12 and 2.14). A more sophisticated example is as follows: the set X := {1/n}n∈ℕ has the same discrete metric topology relative to the regular metric ρ as relative to the discrete metric ρd , which implies that the two metrics are equivalent on X (see Section 2.12.1), i. e., the spaces (X, ρ) and (X, ρd ) and homeomorphic relative the identity mapping Ix := x (see Remarks 2.28). However, the former space is incomplete whereas the latter is complete (see Examples 2.14). Exercise 2.54. (a) Explain. (b) Prove that, if metrics ρ1 and ρ2 on a set X are equivalent in the sense of (2.5), (X, ρ1 ) is complete iff (X, ρ2 ) is complete. 2.13.3 Subspaces of Complete Metric Spaces Proposition 2.4 (Characterization of Completeness). Let (X, ρ) be complete metric space and Y ⊆ X. The subspace (Y, ρ) is complete iff the set Y is closed in (X, ρ). Exercise 2.55. Prove. Exercise 2.56. Apply Proposition 2.4 to show that (a) the spaces ℝ \ {0}, (0, 1), and ℚ are incomplete as subspaces of ℝ, (b) the space (c00 , ρ∞ ) is incomplete as a subspace of (c0 , ρ∞ ), (c) the spaces (c0 , ρ∞ ) and (c, ρ∞ ) are complete as subspaces of l∞ , (d) the space (C[a, b], ρ∞ ) is complete as a subspace of (M[a, b], ρ∞ ), and (e) the space P of all polynomials with real/complex coefficients is incomplete as a subspace of (C[a, b], ρ∞ ). 2.13.4 Nested Balls Theorem The celebrated Nested Intervals Theorem (a form of the completeness of the real numbers) allows the following generalization. Theorem 2.26 (Nested Balls Theorem). A metric space (X, ρ) is complete iff, for every sequence of closed balls ∞ {Bn := B(xn , rn )}n=1 such that Brought to you by  Cambridge University Library Authenticated Download Date  10/11/18 9:53 PM 38  2 Metric Spaces (1) Bn+1 ⊆ Bn , n ∈ ℕ, and (2) rn → 0, n → ∞, the intersection ⋂∞ n=1 Bn is a singleton. Proof. “If” (Sufficiency) part. Under the sufficiency conditions, let {xn }∞ n=1 be an arbitrary fundamental sequence in (X, ρ). Then there exists a subsequence {xn(k) }∞ k=1 (n(k) ∈ ℕ and n(k) < n(k + 1) for k ∈ ℕ) such that ∀ k ∈ ℕ : ρ(xm , xn(k) ) ≤ 1 , m ≥ n(k). 2k+1 (2.8) Exercise 2.57. Explain. Consider the sequence of closed balls {Bk := B(xn(k) , 1/2k )}∞ k=1 . It is nested, i. e., Bk+1 ⊆ Bk , k ∈ ℕ, since, for any y ∈ Bk+1 , by the triangle inequality in view of (2.8), ρ(y, xn(k) ) ≤ ρ(y, xn(k+1) ) + ρ(xn(k+1) , xn(k) ) ≤ 1 2k+1 + 1 2k+1 = 1 , k ∈ ℕ, 2k i. e., y ∈ Bk . As for the radii, we have: 1/2k → 0, k → ∞. Whence, by the premise, we infer that ∞ ∃ x ∈ X : ⋂ Bk = {x} , k=1 and hence, 0 ≤ ρ(xn(k) , x) ≤ 1/2k , k ∈ ℕ, which, by the Squeeze Theorem, implies that xn(k) → x, k → ∞, in (X, ρ). ∞ Since the fundamental sequence {xn }∞ n=1 contains a subsequence {xn(k) }n=1 convergent to x, by the Fundamental Sequence with Convergent Subsequence Proposition (Proposition 2.22) (see Section 2.19, Problem 24), it also converges to x: xn → x, n → ∞, in (X, ρ), which proves the completeness of the space (X, ρ). Brought to you by  Cambridge University Library Authenticated Download Date  10/11/18 9:53 PM 2.13 Completeness and Completion  39 “Only if” (Necessity) part. Suppose that the metric space (X, ρ) is complete and let {Bn := B(xn , rn )}∞ n=1 be a sequence of closed balls satisfying the above conditions. Then the sequence of the centers {xn }∞ n=1 is fundamental in (X, ρ). Indeed, for all n, k ∈ ℕ, since Bn+k ⊆ Bn , 0 ≤ ρ(xn+k , xn ) ≤ rn , which, in view of rn → 0, n → ∞, by the Squeeze Theorem, implies that sup ρ(xn+k , xn ) → 0, n → ∞. k∈ℕ By the completeness of (X, ρ), ∃ x ∈ X : xn → x, n → ∞, in (X, ρ). Let us show that ∞ ⋂ Bn = {x} . n=1 Indeed, since ∀ n ∈ ℕ : xm ∈ Bn , m ≥ n, in view of the closedness of Bn , by the Sequential Characterization of Closed Sets (Theorem 2.18), ∀ n ∈ ℕ : x = lim xm ∈ Bn , m→∞ and hence, we have the inclusion ∞ {x} ⊆ ⋂ Bn . n=1 On the other hand, for any y ∈ ⋂∞ n=1 Bn , since x, y ∈ Bn for each n ∈ ℕ, by the triangle inequality, 0 ≤ ρ(x, y) ≤ ρ(x, xn ) + ρ(xn , y) ≤ 2rn , which, in view of rn → 0, n → ∞, by the Squeeze Theorem, implies that ρ(x, y) = 0, and hence, by the separation axiom, x = y. Brought to you by  Cambridge University Library Authenticated Download Date  10/11/18 9:53 PM 40  2 Metric Spaces Thus, the inverse inclusion ∞ {x} ⊇ ⋂ Bn . n=1 holds as well, and we conclude that ∞ ⋂ Bn = {x}. n=1 Remark 2.34. Each of the four necessity conditions in the Nested Balls Theorem (the completeness of the space, the closedness of the balls, and conditions (1) and (2)) is essential and cannot be relaxed or dropped. Exercise 2.58. Verify by providing corresponding counterexamples. A more general version of the Nested Balls Theorem, which can be proved by mimicking the proof of the latter, is the following Theorem 2.27 (Generalized Nested Balls Theorem). A metric space (X, ρ) is complete iff for every sequence of nonempty closed sets {Bn }∞ n=1 such that (1) Bn+1 ⊆ Bn , n ∈ ℕ, and (2) diam(Bn ) := supx,y∈Bn ρ(x, y) → 0, n → ∞, the intersection ⋂∞ n=1 Bn is a singleton. 2.13.5 Completion The following beautiful construct of obtaining a complete metric space from an arbitrary one is crucial in functional analysis. Theorem 2.28 (Completion Theorem for Metric Spaces). An arbitrary metric space (X, ρ) can be isometrically embedded as a dense subspace in a complete metric space (X,̃ ρ)̃ called a completion of (X, ρ). Any two completions of (X, ρ) are isometric. Proof. On the set 𝒳 of all Cauchy sequences in (X, ρ), let us define the asymptotic equivalence relation as follows: ∞ {xn }∞ n=1 ∼ {yn }n=1 ⇔ ρ(xn , yn ) → 0, n → ∞. Exercise 2.59. Verify the equivalence axioms. The collection of all equivalence classes X̃ is a metric space relative to the mapping ̃ X̃ ∋ [x], [y] → ρ([x], [y]) := lim ρ(xn , yn ), n→∞ Brought to you by  Cambridge University Library Authenticated Download Date  10/11/18 9:53 PM (2.9) 2.13 Completeness and Completion  41 ∞ where x := {xn }∞ n=1 and y := {yn }n=1 are arbitrary representatives of the classes [x] and [y], respectively. Exercise 2.60. (a) Use the Quadrilateral Inequality (Theorem 2.8) to prove that the mapping ρ̃ is well defined, i. e., the limit exists and is independent of the choice of the representatives of the equivalence classes. (b) Prove that ρ̃ is a metric on X.̃ ̃ ̃ The space (X,̃ ρ)̃ is complete. Indeed, let {[xn ]}∞ n=1 be a Cauchy sequence in (X, ρ), (n) ∞ with xn := {xk }k=1 being a representative of the class [xn ], n ∈ ℕ. Since for each n ∈ ℕ, the sequence {xk(n) }∞ k=1 is fundamental in (X, ρ), (n) ∀ n ∈ ℕ ∃ k(n) ∈ ℕ ∀ p ≥ k(n) : ρ (xp(n) , xk(n) )< 1 . n (2.10) (n) ∞ Let show that the sequence {xk(n) }n=1 is fundamental in (X, ρ). For any m, n, p ∈ ℕ, by the triangle inequality, (m) (n) (m) (n) ρ (xk(m) , xk(n) ) ≤ ρ (xk(m) , xp(m) ) + ρ (xp(m) , xp(n) ) + ρ (xp(n) , xk(n) ). (2.11) ̃ ̃ Since the sequence {[xn ]}∞ n=1 is fundamental in (X, ρ), ε ̃ m ], [xn ]) = lim ρ (xp(m) , xp(n) ) < , ∀ ε > 0 ∃N ∈ ℕ ∀ m, n ≥ N : ρ([x p→∞ 2 and hence, ∀ ε > 0 ∃N ∈ ℕ ∀ m, n ≥ N ∃ K(m, n) ∈ ℕ ∀ p ≥ K(m, n) : ε ρ (xp(m) , xp(n) ) < . 2 (2.12) Without loss of generality, we can regard N ∈ ℕ to be large enough so that 1 ε < , n ≥ N. n 4 Let us fix arbitrary m, n ≥ N. Then for all p ≥ max [k(m), k(n), K(m, n)] , in view of the choice of k(m) and k(n) (see (2.10)), we have: (m) ρ (xk(m) , xp(m) ) < 1 ε < m 4 (n) )< and ρ (xp(n) , xk(n) 1 ε < . n 4 (2.13) From (2.11)–(2.13), we infer that (m) (n) ρ (xk(m) , xk(n) ) < ε, m, n ≥ N, (n) ∞ and hence, the sequence {xk(n) }n=1 is fundamental in (X, ρ). Brought to you by  Cambridge University Library Authenticated Download Date  10/11/18 9:53 PM 42  2 Metric Spaces ̃ ̃ to the equivalence Let us show now that the sequence {[xn ]}∞ n=1 converges in (X, ρ) (n) ∞ class [x] represented by the sequence x := {xk(n) }n=1 . Indeed, for any fixed n ∈ ℕ, (p) ̃ n ], [x]) = lim ρ (xp(n) , xk(p) ρ([x ) by the triangle inequality; p→∞ (p) (n) (n) ≤ lim ρ (xp(n) , xk(n) ) + lim ρ (xk(n) , xk(p) ) p→∞ ≤ by the choice of k(n) (see (2.10)); p→∞ 1 (p) (n) + lim ρ (xk(n) ). , xk(p) n p→∞ (2.14) (n) ∞ Since the sequence {xk(n) }n=1 is fundamental in (X, ρ), ε (p) (n) ∀ ε > 0 ∃ N ∈ ℕ ∀ n, p ≥ N : ρ (xk(n) , xk(p) )< . 2 For each fixed n ≥ N, passing to the limit as p → ∞, we arrive at ε (p) (n) lim ρ (xk(n) , xk(p) ) ≤ , n ≥ N. 2 p→∞ (2.15) Without loss of generality, we can regard N ∈ ℕ to be large enough so that 1 ε < , n ≥ N. n 2 Considering this, we infer from inequalities (2.14) and (2.15) that ̃ n ], [x]) < ε, n ≥ N, ρ([x and hence, ̃ lim [xn ] = [x] in (X,̃ ρ), n→∞ ̃ which proves the completeness of (X,̃ ρ). The isometric embedding T : X → X̃ is obtained by associating with each x ∈ X the equivalence class Tx ∈ X̃ represented by the constant sequence {x, x, . . . , x, . . . }, and hence, consisting of all sequences convergent to x in (X, ρ). Exercise 2.61. Verify. Thus, T(X) is the set of all equivalence classes of sequences convergent in (X, ρ). To show that T(X) is dense in X, consider an arbitrary equivalence class [x] ∈ X̃ represented by a Cauchy sequence {xn }∞ n=1 . Since ∀ ε > 0 ∃ N ∈ ℕ ∀ m, n ≥ N : ρ(xm , xn ) < ε, we have: ∀ n ≥ N : ρ̃ ([x], Txn ) = lim ρ(xm , xn ) ≤ ε, m→∞ Brought to you by  Cambridge University Library Authenticated Download Date  10/11/18 9:53 PM 2.14 Category and Baire Category Theorem  43 i. e., ̃ lim Txn = [x] in (X,̃ ρ), n→∞ which, by the Sequential Characterization of Dense Sets (Theorem 2.20), implies that T(X) = X.̃ Thus, (X,̃ ρ)̃ is a completion of (X, ρ). It remains now to prove the uniqueness of completion up to an isometry. Observe that a completion (X,̃ ρ)̃ of (X, ρ) encompasses a complete metric space ̃ ̃ i. e., forms a triple (X, ρ)̃ along with an isometric dense embedding T of (X, ρ) in (X,̃ ρ), (X,̃ ρ,̃ T). Let (X,̃ ρ,̃ T) and (Y,̃ σ,̃ S) be two arbitrary completions of (X, ρ). We obtain an isometry of X̃ onto Ỹ by continuously extending the isometry ST −1 of T(X), which is dense in X,̃ onto S(X), which is dense in Y.̃ Exercise 2.62. Describe the extension process. Remark 2.35. It follows immediately that, if X0 is a dense subset of a complete metric space (X, ρ), the latter is a completion of the space (X0 , ρ). Examples 2.15. 1. Since ℚ = ℝ, ℝ is a completion of ℚ. 2. More generally, since ℚn is a dense subspace of the real lp(n) (n ∈ ℕ, 1 ≤ p ≤ ∞), the latter is a completion of (ℚn , ρp ). Remark 2.36. The complex counterpart is obvious. 3. Since the set c00 is dense in the complete space (c0 , ρ∞ ) (see Exercise 2.56), the latter is a completion of (c00 , ρ∞ ). The same is true for the pair lp (1 ≤ p < ∞) and (c00 , ρp ). Exercise 2.63. Verify. 4. Since the set P of all polynomials with real/complex coefficients is dense subspace of (C[a, b], ρ∞ ), the latter is a completion of (P, ρ∞ ). Exercise 2.64. Construct completions of ℝ \ {0} and (0, 1) relative to the usual metric. 2.14 Category and Baire Category Theorem 2.14.1 Nowhere Denseness First, we are to introduce and study nowhere dense sets, i. e., sets in a metric space, which are, in a certain sense, “scarce”. Brought to you by  Cambridge University Library Authenticated Download Date  10/11/18 9:53 PM 44  2 Metric Spaces Definition 2.30 (Nowhere Dense Set). A set A in a metric space (X, ρ) is called nowhere dense if the interior of its closure A is empty: int(A) = 0, i. e., A contains no nontrivial open balls. Remarks 2.37. – A set A is nowhere dense in a metric space (X, ρ) iff its closure A is nowhere dense in (X, ρ) (see Section 2.19, Problem 15). – For a closed set A in a metric space (X, ρ), since A = A, its nowhere denseness is simply the emptiness of its interior: int(A) = 0. Examples 2.16. 1. The empty set 0 is nowhere dense in any metric space (X, ρ) and only the empty set is nowhere dense in a discrete metric space (X, ρd ). 2. Finite sets, the sets ℤ of all integers and {1/n}n∈ℕ are nowhere dense in ℝ. 3. The celebrated Cantor set (see, e. g., [21, 23, 24]) is a closed nowhere dense set in ℝ as well as its twodimensional analogue, the Sierpinski7 carpet, in ℝ2 with the Euclidean distance (i. e., in l2(2) (ℝ)) (see, e. g., [38]). 4. The set ℝ of the reals is nowhere dense in l2(2) (ℝ). 5. An arbitrary dense set in a metric space (X, ρ) is not nowhere dense. In particular, the sets ℚ of the rationals and the set ℚc of the irrationals are not nowhere dense in ℝ. 6. However, a set need not be dense in a metric space not to be nowhere dense. Thus, an nontrivial proper interval I in ℝ is neither dense nor nowhere dense. Exercise 2.65. Verify. Remark 2.38. Formulated entirely in terms of closure and interior, nowhere denseness is a topological property, i. e., it is preserved by a homeomorphism. Proposition 2.5 (Characterization of Nowhere Denseness). A set A is nowhere dense in a metric space (X, ρ) iff its exterior ext(A) is dense in (X, ρ). Proof. In view of the fact that, for any B ⊆ X, ext(B) = X \ B 7 Waclaw Sierpinski (1882–1969). Brought to you by  Cambridge University Library Authenticated Download Date  10/11/18 9:53 PM 2.14 Category and Baire Category Theorem  45 (see Section 2.19, Problem 21), the statement immediately follows from the following disjoint union representation (first, setting B := X \ A, then setting B := A): X = ext(X \ A) ∪ X \ A = int(A) ∪ ext(A). Exercise 2.66. Explain. Since for a closed set A in a metric space (X, ρ), ext(A) = Ac (see Section 2.19, Problem 21), we immediately obtain the following Corollary 2.2 (Characterization of Nowhere Denseness of Closed Sets). A closed set A in a metric space (X, ρ) is nowhere dense iff its open complement Ac is dense in (X, ρ). By the complementary nature of closed and open sets (see Theorem 2.18), we can equivalently reformulate the prior statement as follows: Corollary 2.3 (Characterization of Denseness of Open Sets). An open set A in a metric space (X, ρ) is dense iff its closed complement Ac is nowhere dense in (X, ρ). Example 2.17. The complement of the Cantor set is an open dense set in ℝ. Theorem 2.29 (Properties of Nowhere Dense Sets). The nowhere dense sets in a metric space (X, ρ) have the following properties: (1) a subset of a nowhere dense set is nowhere dense; (2) an arbitrary intersection of nowhere dense sets is nowhere dense; (3) a finite union of nowhere dense sets is nowhere dense. Exercise 2.67. Prove (cf. Properties of Bounded Sets (Theorem 2.13)). Hint. To prove (3), first show that, for any sets A and B in (X, ρ), A ∪ B = A ∪ B, which is equivalent to ext(A ∪ B) = ext(A) ∩ ext(B) (see Section 2.19, Problem 21), and then exploit the denseness jointly with openness (cf. Finite Intersections of Open Dense Sets Proposition (Proposition 2.24) (Section 2.19, Problem 29)). Remark 2.39. An infinite union of nowhere dense sets in a metric space (X, ρ) need not be nowhere dense. For instance, any singleton is nowhere dense in ℝ. However, ℚ, being a countably infinite union of singletons, is dense in ℝ. Brought to you by  Cambridge University Library Authenticated Download Date  10/11/18 9:53 PM 46  2 Metric Spaces 2.14.2 Category Here, based on the concept of nowhere denseness, we quite naturally divide the sets of a metric space into two categories, category giving a sense of a set’s “fullness” and being closely related to the notion of completeness. Definition 2.31 (First/SecondCategory Set). A set A in a metric space (X, ρ) is said to be of the first category if it can be represented as a countable union of nowhere dense sets. Otherwise, A is said to be of the second category in (X, ρ). Examples 2.18. 1. A nowhere dense set in a metric space (X, ρ) is a firstcategory set. In particular, the set ℕ of all naturals is of the first category in ℝ. 2. A dense set in a metric space (X, ρ) may be of the firstcategory as well. For instance, the set ℚ of all rationals in ℝ. 3. The space (c00 , ρ∞ ) is of the first category in itself (also in (c0 , ρ∞ ), (c, ρ∞ ), and l∞ ). Indeed, ∞ c00 = ⋃ Un , n=1 where Un := {x := {x1 , . . . , xn , 0, 0, . . . }} , n ∈ ℕ. (n) Each Un is closed, being an isometric embedding of the complete space (l∞ , ρ∞ ) into (c00 , ρ∞ ), and nowhere dense in (c00 , ρ∞ ) (also in (c0 , ρ∞ ), (c, ρ∞ ), and l∞ ) since ∀ n ∈ ℕ ∀ x := {x1 , . . . , xn , 0, 0, . . . } ∈ Un ∀ ε > 0 ∃y := {x1 , . . . , xn , ε/2, 0, 0 . . . } ∈ c00 \ Un : ρ∞ (x, y) = ε/2 < ε, which implies that int(Un ) = 0, n ∈ ℕ. 4. As follows from the Baire8 Category Theorem (Theorem 2.31), every complete metric space (X, ρ) is of the second category in itself. In particular, the complete spaces ℝ and ℂ are of the second category in themselves, the former being of the first category (nowhere dense, to be precise) in the latter (cf. Examples 2.16). 5. Every nonempty set in a discrete metric space (X, ρd ) is a secondcategory set. 8 RenéLouis Baire (1874–1932). Brought to you by  Cambridge University Library Authenticated Download Date  10/11/18 9:53 PM 2.14 Category and Baire Category Theorem  47 Exercise 2.68. Explain 5. Remark 2.40. Formulated entirely in terms of nowhere denseness and union, first category is a topological property, i. e., it is preserved by a homeomorphism, and hence, so is second category. Example 2.19. Thus, every open interval I in ℝ, being homeomorphic to ℝ (see Remark 2.29), is of the second category in itself as a subspace of ℝ and, since the interior of a set in I coincides with its interior in ℝ, is also of the second category in ℝ. Theorem 2.30 (Properties of FirstCategory Sets). The firstcategory sets in a metric space (X, ρ) have the following properties: (1) a subset of a firstcategory set is a firstcategory set; (2) an arbitrary intersection of firstcategory sets is a firstcategory set; (3) an arbitrary countable union of firstcategory sets is a firstcategory set. Exercise 2.69. Prove. We immediately obtain the following Corollary 2.4 (Set With a SecondCategory Subset). A set A in a metric space (X, ρ) containing a secondcategory subset B is of the second category. Examples 2.20. 1. Every nontrivial (open or closed) interval I in ℝ is of the second category in ℝ and in itself as a subspace of ℝ since it contains an open interval, which is a secondcategory set in ℝ (see Example 2.19). 2. More generally, for the same reason, any set A ⊆ ℝ with int(A) ≠ 0 is of the second category in ℝ and in itself as a subspace of ℝ. 2.14.3 Baire Category Theorem The Baire Category Theorem, already referred to in the prior section, is one of the most important facts about complete metric spaces critical for proving a number of fundamental statements such as the Uniform Boundedness Principle (Theorem 6.7) and the Open Mapping Theorem (Theorem 6.13). We are to prove the celebrated statement now. Theorem 2.31 (Baire Category Theorem). A complete metric space (X, ρ) is of the second category in itself. Proof. Let us prove the statement by contradiction assuming that there is a complete metric space (X, ρ) of the first category in itself, i. e., X can be represented as a countable union of nowhere dense sets: ∞ X = ⋃ Un . n=1 Brought to you by  Cambridge University Library Authenticated Download Date  10/11/18 9:53 PM 48  2 Metric Spaces Without loss of generality, we can regard all Un , n ∈ ℕ, to be closed. Exercise 2.70. Explain. Being nowhere dense, the closed set U1 is a proper subset of X, and hence, by the Characterization of Nowhere Denseness of Closed Sets (Corollary 2.2), its open complement U1c is dense in (X, ρ), and the more so, nonempty. Therefore, ∃ x1 ∈ U1c ∃ 0 < ε1 < 1 : B(x1 , ε1 ) ⊆ U1c , i. e., B(x1 , ε1 ) ∩ U1 = 0, Since the open ball B(x1 , ε1 /2) is not contained in the closed nowhere dense set U2 , ∃ x2 ∈ B(x1 , ε1 /2) ∃ 0 < ε2 < 1/2 : B(x2 , ε2 ) ∩ U2 = 0 and B(x2 , ε2 ) ⊆ B(x1 , ε1 /2). Continuing inductively, we obtain a sequence of closed balls {B(xn , εn )}∞ n=1 such that (1) B(xn+1 , εn+1 ) ⊆ B(xn , εn ), n ∈ ℕ. (2) 0 < εn < 1/2n−1 , n ∈ ℕ. (3) B(xn , εn ) ∩ Un = 0, n ∈ ℕ. From (1) and (2), by the Nested Balls Theorem (Theorem 2.26), we infer that ∞ ⋂ B(xn , εn ) = {x} n=1 with some x ∈ X. Since x ∈ B(xn , εn ) for each n ∈ ℕ, by (3), we conclude that ∞ x ∈ ̸ ⋃ Un = X, n=1 which is a contradiction proving the statement. Examples 2.21. 1. By the Baire Category Theorem, the complete spaces ℝ and ℂ are of the second category in themselves (see Examples 2.18). 2. Any set A ⊆ ℝ with int(A) ≠ 0 is of the second category in ℝ and in itself as a subspace of ℝ (cf. Examples 2.20). Brought to you by  Cambridge University Library Authenticated Download Date  10/11/18 9:53 PM 2.15 Compactness  49 3. However, a set need not have a nonempty interior to be of the second category. Indeed, as follows from the Baire Category Theorem, the set ℚc of all irrationals with int(ℚc ) = 0 is of the second category in ℝ, as well as the complement Ac of any firstcategory set A in a complete metric space (X, ρ) (see Section 2.19, Problem 31). 4. By the Baire Category Theorem and the Characterization of Completeness (Proposition 2.4), the set of all integers ℤ and the Cantor set are of the second category in themselves as closed subspaces of the complete space ℝ, but are of the first category (nowhere dense, to be precises) in ℝ (cf. Examples 2.16). Remarks 2.41. – The converse to the Baire Category Theorem is not true, i. e., there exist incomplete metric spaces of the second category in themselves. For instance, an open interval I in ℝ is of the second category in itself (see Example 2.19), but is incomplete as a subspace of ℝ. – The proof of the Baire Category Theorem requires a weaker form of the Axiom of Choice (see Appendix A), the Axiom of Dependent Choices (see, e. g., [22, 36]). Corollary 2.5 (SecondCategory Properties of Complete Metric Spaces). In a complete metric space (X, ρ), (1) any representation of X as a countable union of its subsets ∞ X = ⋃ Un n=1 contains at least one subset UN , which is not nowhere dense, i. e., ∃ N ∈ ℕ : int(UN ) ≠ 0; (2) any countable intersection ⋂∞ n=1 Un of open dense sets is nonempty, i. e., ∞ ⋂ Un ≠ 0. n=1 Exercise 2.71. Prove. Hint. To prove (2) use the Characterization of Nowhere Denseness of Closed Sets (Corollary 2.2) (cf. the Finite Intersections of Open Dense Sets Proposition (Proposition 2.24)). 2.15 Compactness The notion of compactness, naturally emerging from our spatial intuition, is of utmost importance both in theory and applications. We are to study it and related concepts here. Brought to you by  Cambridge University Library Authenticated Download Date  10/11/18 9:53 PM 50  2 Metric Spaces 2.15.1 Total Boundedness 2.15.1.1 Definitions and Examples Total boundedness is a notion inherent to metric spaces that is stronger than boundedness, but weaker than precompactness. Definition 2.32 (εNet). Let (X, ρ) be a metric space and ε > 0. A set Nε ⊆ X is called an εnet for a set A ⊆ X if A can be covered by the collection {B(x, ε)  x ∈ Nε } of all open εballs centered at the points of Nε , i. e., A ⊆ ⋃ B(x, ε). x∈Nε Examples 2.22. 1. For an arbitrary nonempty set A in a metric space (X, ρ) and any ε > 0, A is an εnet for itself. 2. A dense set A in a metric space (X, ρ) is an εnet for the entire X with any ε > 0. In particular, for any ε > 0, ℚ and ℚc are εnets for ℝ. 3. For any ε > 0, ℚ is an εnet for ℚc . 4. For any n ∈ ℕ and ε > 0, the set Nε := {( εk εk1 , . . . , n ) ki ∈ ℤ, i = 1, . . . , n} √n √n is an εnet for ℝn with the Euclidean metric, i. e., for l2(n) (ℝ). Exercise 2.72. Verify and make a drawing for n = 2. Remark 2.42. Example 4 can be modified to furnish an εnet for ℝn endowed with any pmetric (1 ≤ p ≤ ∞), i. e., for lp(n) (ℝ) and can be naturally stretched to the complex nspace ℂn relative to any pmetrics (1 ≤ p ≤ ∞), i. e., to any space lp(n) (ℂ). Exercise 2.73. Verify for n = 2, p = 1 and for n = 2, p = ∞. Remarks 2.43. As the prior examples demonstrate, an εnet Nε for a set A in a metric space (X, ρ) need not consist of points of A. It may even happen that A and Nε are disjoint. Definition 2.33 (Total Boundedness). A set A in a metric space (X, ρ) is called totally bounded if, for any ε > 0, there is a finite εnet for A: N ∀ ε > 0 ∃ N ∈ ℕ, ∃ {x1 , . . . , xN } ⊆ X : A ⊆ ⋃ B(xn , ε), n=1 i. e., A can be covered by a finite number of εballs, however small their radii. A metric space (X, ρ) is called totally bounded if the set X is totally bounded in (X, ρ). Brought to you by  Cambridge University Library Authenticated Download Date  10/11/18 9:53 PM 2.15 Compactness  51 Remarks 2.44. – A set A is totally bounded in a metric space (X, ρ) iff totally bounded is its closure A. – For a totally bounded set A in a metric space (X, ρ) and any ε > 0, a finite εnet for A can be chosen to consist entirely of points of A. – The total boundedness of a nonempty set A in a metric space (X, ρ) is equivalent to the total boundedness of (A, ρ) as a subspace of (X, ρ). Exercise 2.74. Verify. Hint. Consider a finite ε/2net {x1 , . . . , xN } ⊆ X (N ∈ ℕ) for A and construct an εnet {y1 , . . . , yN } ⊆ A. Examples 2.23. 1. Finite sets, including the empty set 0, are totally bounded in an arbitrary metric space (X, ρ) and only finite sets are totally bounded in a discrete metric space (X, ρd ). 2. A bounded set A in the nspace ℝn (n ∈ ℕ) with the Euclidean metric, i. e., in l2(n) (ℝ), is totally bounded. Indeed, being bounded, the set A is contained in a hypercube Jm = [−m, m]n with some m ∈ ℕ. Then, for any ε > 0, Nε := Nε ∩ Jm , where Nε is the εnet for l2(n) (ℝ) from Examples 2.22, is a finite εnet for A. Remark 2.45. The same is true for any (real or complex) lp(n) (1 ≤ p ≤ ∞) (see Remark 2.42). 3. In particular, (0, 1] and {1/n}n∈ℕ are totally bounded sets in ℝ. The set E := {en := {δnk }∞ k=1 }n∈ℕ , where δnk is the Kronecker delta, is bounded, but not totally bounded in (c00 , ρ∞ ) (also in (c0 , ρ∞ ), (c, ρ∞ ), and l∞ ) since there is no finite 1/2net for A. The same example works in lp (1 ≤ p < ∞). Exercise 2.75. Verify. 2.15.1.2 Properties Theorem 2.32 (Properties of Totally Bounded Sets). The totally bounded sets in a metric space (X, ρ) have the following properties: (1) a totally bounded set is necessarily bounded, but not vice versa; (2) a subset of a totally bounded set is totally bounded; (3) an arbitrary intersection of totally bounded sets is totally bounded; (4) a finite union of totally bounded sets is totally bounded. Brought to you by  Cambridge University Library Authenticated Download Date  10/11/18 9:53 PM 52  2 Metric Spaces Exercise 2.76. (a) Prove. (b) Give an example showing that an infinite union of totally bounded sets need not be totally bounded. Exercise 2.77. Using the set E from Examples 2.23, show that a nontrivial sphere/ball in (c00 , ρ∞ ) (also in (c0 , ρ∞ ), (c, ρ∞ ), and lp (1 ≤ p ≤ ∞)) is not totally bounded. From the prior proposition and Examples 2.23, we obtain Corollary 2.6 (Characterization of Total Boundedness in the nSpace). A set A is totally bounded in the (real or complex) space lp(n) (n ∈ ℕ, 1 ≤ p ≤ ∞) iff it is bounded. It is remarkable that total boundedness in a metric space can be characterized in terms of fundamentality as follows. Theorem 2.33 (Characterization of Total Boundedness). A nonempty set A is totally bounded in a metric space (X, ρ) iff every sequence {xn }∞ n=1 in A contains a fundamental subsequence {xn(k) }∞ . k=1 Proof. “Only if” part. Suppose a set A is totally bounded in a metric space (X, ρ) and let {xn }∞ n=1 be an arbitrary sequence in A. If the set A is finite, i. e., A = {y1 , . . . , yN } with some N ∈ ℕ, then {xn }∞ n=1 necessarily assumes the same value yi with some i = 1, . . . , N for infinitely many indices n ∈ ℕ (i. e., “frequently”). Exercise 2.78. Explain. Hence, {xn }∞ n=1 contains a constant subsequence, which is fundamental. Now, assume that the set A is infinite and let {xn }∞ n=1 be an arbitrary sequence in A. If {xn }∞ assumes only a finite number of distinct values, we arrive at the prior case. n=1 Suppose that {xn }∞ assumes infinite many distinct values. Then, without loss of n=1 generality, we can regard that xm ≠ xn , m, n ∈ ℕ. By the total boundedness of A, since it is coverable by a finite number of 1balls, there must exist a 1ball B(y1 , 1) with some y1 ∈ X, which contains infinitely many terms ∞ ∞ of {xn }∞ n=1 , and hence, a subsequence {x1,n }n=1 of {xn }n=1 . Similarly, there must exist a 1/2ball B(y2 , 1/2) with some y2 ∈ X, which contains ∞ ∞ infinitely many terms of {x1,n }∞ n=1 , and hence, a subsequence {x2,n }n=1 of {x1,n }n=1 . Continuing inductively, we obtain a countable collection of sequences ∞ {{xm,n }n=1 m ∈ ℤ+ } , ∞ such that, for each m ∈ ℕ, {xm,n }∞ n=1 is a subsequence of {x(m−1),n }n=1 , with ∞ {x0,n }n=1 := {xn }∞ n=1 , Brought to you by  Cambridge University Library Authenticated Download Date  10/11/18 9:53 PM 2.15 Compactness  53 and ∞ {xm,n }n=1 ⊆ B(ym , 1/m) with some ym ∈ X. ∞ Then the “diagonal subsequence” {xn,n }∞ n=1 is a fundamental subsequence of {xn }n=1 since, by the triangle inequality, ∀ n ∈ ℕ, ∀ m ≥ n : ρ(xm,m , xn,n ) ≤ ρ(xm,m , yn ) + ρ(yn , xn,n ) < 1/n + 1/n = 2/n. “If” part. Let us prove this part by contrapositive assuming that a set A is not totally bounded in a metric space (X, ρ). Then there is an ε > 0 such that there does not exist a finite εnet for A. In particular, for an arbitrary x1 ∈ A the εball B(x1 , ε) does not cover A, and hence, ∃ x2 ∈ A : x2 ∉ B(x1 , ε). Similarly, the εballs B(x1 , ε) and B(x2 , ε) do not cover A, and hence, ∃ x3 ∈ A : x3 ∉ B(x1 , ε) ∪ B(x2 , ε). Continuing inductively, we obtain a sequence {xn }∞ n=1 of points of A such that n−1 xn ∉ ⋃ B(xk , ε), n ≥ 2, k=1 i. e., all of wh