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Marat V. Markin
Elementary Functional Analysis

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Marat V. Markin

Elementary
Functional
Analysis
|

Mathematics Subject Classification 2010
46-02, 47-02, 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 978-3-11-061391-9
e-ISBN (PDF) 978-3-11-061403-9
e-ISBN (EPUB) 978-3-11-061409-1
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 nineteenth-century 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 “tell-it-all” 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, one-semester graduate curriculum
(fifteen weeks with two seventy-five-minute 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/10.1515/9783110614039-201

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
Hahn-Banach 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 Well-Ordering 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 n-Tuples | 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
Finite-Dimensional Spaces and Related Topics | 125
Norm Equivalence and Completeness | 125
Finite-Dimensional 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
Rank-Nullity 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
Self-Duality of Hilbert Spaces | 261
Riesz Representation Theorem | 261
Linear Bounded Functionals on Certain Hilbert Spaces | 264
Weak Convergence in Hilbert Spaces | 264
Duality of Finite-Dimensional Spaces | 265
Representation Theorem | 265
Weak Convergence in Finite-Dimensional 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 n-spaces of all ordered n-tuples 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/10.1515/9783110614039-001

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
one-to-one 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 Schröder-Bernstein 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 three-dimensional 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/10.1515/9783110614039-002

8 | 2 Metric Spaces
3.

Let n ∈ ℕ and 1 ≤ p ≤ ∞. The real/complex n-space ℝn or ℂn is a metric space
relative to the p-metric
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

(p-summable/bounded sequences, respectively) is a metric space relative to the
p-metric
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 n-tuples
(n ∈ ℕ) and sequences. We use Hölder’s inequality to prove Minkowski’s inequality for
the case of n-tuples. In turn, to prove Hölder’s inequality for n-tuples, 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 n-tuples.
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 n-Tuples
Definition 2.3 (p-Norm of an n-Tuple). Let n ∈ ℕ. For an n-tuple x := (x1 , . . . , xn ) ∈ ℂn
(1 ≤ p ≤ ∞), the p-norm of x is the distance of x from the zero n-tuple 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 n-Tuples). 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 n-Tuples). 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 (p-Norm 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 n-tuples,
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 n-Tuples (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 n-tuples 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 p-metric
(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 set-theoretic 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/complex-valued functions
bounded on T, i. e., all functions f : T → ℝ (or ℂ) such that
sup |f (t)| < ∞,
t∈T

4 Augustin-Louis 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/complex-valued 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/complex-valued 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 real-valued 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 ε-ball 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.

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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.

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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 ).

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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 → ∞.

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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.

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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 n-tuples
with (real/complex) rational components.
2. The spaces lp are separable for 1 ≤ p < ∞.

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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.

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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 n-space ℝn or ℂn (n ∈ ℕ), all p-metrics (1 ≤ p ≤ ∞) (see Examples 2.1) are equivalent in the sense of (2.5).
Hint. Show the equivalence of any p-metric ρp (1 ≤ p < ∞) in the sense of (2.5) to
ρ∞ .

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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

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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 one-to-one (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.

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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.

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2.13 Completeness and Completion

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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 n-Space). The (real or complex) space lp(n)
(n ∈ ℕ, 1 ≤ p ≤ ∞) is complete.
Exercise 2.51. Prove.
Hint. Considering the equivalence of all p-metrics on the n-space in the sense of
(2.5) (see Exercise 2.43) and Exercise 2.54, it suffices to show the completeness of
the n-space 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 → ∞.

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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 → ∞.

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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.

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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.

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2.13 Completeness and Completion

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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

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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).

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2.13 Completeness and Completion

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“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.

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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→∞

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(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).

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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→∞

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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”.

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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 two-dimensional 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).

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2.14 Category and Baire Category Theorem

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(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.

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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/Second-Category 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 first-category 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 first-category 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 second-category set.
8 René-Louis Baire (1874–1932).

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2.14 Category and Baire Category Theorem

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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 First-Category Sets). The first-category sets in a metric
space (X, ρ) have the following properties:
(1) a subset of a first-category set is a first-category set;
(2) an arbitrary intersection of first-category sets is a first-category set;
(3) an arbitrary countable union of first-category sets is a first-category set.
Exercise 2.69. Prove.
We immediately obtain the following
Corollary 2.4 (Set With a Second-Category Subset). A set A in a metric space (X, ρ)
containing a second-category 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

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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).

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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
first-category 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 (Second-Category 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.

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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 (ε-Net). Let (X, ρ) be a metric space and ε > 0. A set Nε ⊆ X is called
an ε-net for a set A ⊆ X if A can be covered by the collection {B(x, ε) | x ∈ Nε } of all open
ε-balls 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
ε-net for itself.
2. A dense set A in a metric space (X, ρ) is an ε-net for the entire X with any ε > 0.
In particular, for any ε > 0, ℚ and ℚc are ε-nets for ℝ.
3. For any ε > 0, ℚ is an ε-net 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 ε-net 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 ε-net for ℝn endowed with any
p-metric (1 ≤ p ≤ ∞), i. e., for lp(n) (ℝ) and can be naturally stretched to the complex
n-space ℂn relative to any p-metrics (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 ε-net 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 ε-net 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 ε-balls, however small their radii.
A metric space (X, ρ) is called totally bounded if the set X is totally bounded in
(X, ρ).

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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 ε-net 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/2-net {x1 , . . . , xN } ⊆ X (N ∈ ℕ) for A and construct an ε-net
{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 n-space 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 ε-net for
l2(n) (ℝ) from Examples 2.22, is a finite ε-net 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 1/2-net 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.

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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 n-Space). 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 1-balls,
there must exist a 1-ball 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 1/2-ball 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 ,

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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 ε-net for A.
In particular, for an arbitrary x1 ∈ A the ε-ball B(x1 , ε) does not cover A, and hence,
∃ x2 ∈ A : x2 ∉ B(x1 , ε).
Similarly, the ε-balls 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