[problem books in mathematics] exercises in analysis ||
TRANSCRIPT
Problem Books in Mathematics
Series editor:
Peter Winkler Department of Mathematics Dartmouth College Hanover, NH 03755 USA
For further volumes: http://www.springer.com/series/714
Exercises in Analysis
and Computer Science Jagiellonian University Krakow, Poland
Nikolaos S. Papageorgiou Department of Mathematics National Technical University Athens, Greece
ISSN 0941-3502 ISBN 978-3-319-06175-7 ISBN 978-3-319-06176-4 (eBook) DOI 10.1007/978-3-319-06176-4 Springer Cham Heidelberg New York Dordrecht London
Library of Congress Control Number: 2014940530
Mathematics Subject Classification (2010): 00A07, 46-XX, 30LXX, 28-XX, 60G46
© Springer International Publishing Switzerland 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publica- tion does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of pub- lication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.
Printed on acid-free paper
The aim of this book is to review the theory of some basic topics in Analysis and accompany the theory with problems and their solutions. With the problems the reader can test his/her understanding of the theory and also discover extensions of the theory and additional results which are not so standard in the literature. The topics covered span a more or less standard advanced undergraduate and graduate curricu- lum in Analysis. More precisely, we focus on the following subjects:
1. Metric Spaces
2. Topological Spaces (here there is also some introductory material on Algebraic Topology)
3. Measure, Integration and Martingales (including Lp-spaces)
4. Measure and Topology (covering issues concerning the interplay between measure theory and topology)
5. Functional Analysis (with emphasis on basic Banach space theory).
Each one of the above five subjects corresponds to a different chap- ter. In the first part of each chapter, we present the basic theory, with all the main definitions and results. We also include comments and remarks expanding on the concepts and results, but no proofs. This review material will help the reader refresh his/her knowledge of the theory before tackling the problems. In each chapter the theory is fol- lowed by problems and their detailed solution. In each chapter, there are at least 170 problems, marked with , or according to their difficulty. Some of those problems complement the theory, while the rest check the reader’s understanding of the theory. We strongly encourage the reader to put some substantial personal effort in trying to solve the problem, before looking at the solution. Otherwise, they will get no benefit from reading the book. On the other hand, a serious
v
vi Preface
effort to solve the problem themselves and subsequently compare their solution with the one provided (or checking to see where they failed to come up with the right arguments to produce a solution) will help them to achieve a solid understanding of the theory.
It is not easy to provide the origin of each of the problems and of their solutions. They can be traced as problems included in the books mentioned in the literature (where they are stated without proofs), or they can be found in the problem books listed in the References, or they are standard exercises in the public domain, or they were accumulated through the years from teaching undergraduate and graduate courses on these or closely related subjects. In any case, the books mentioned in the References can be a valuable source for additional theoretical material and more problems. Our book is only the starting point (helpful we hope).
The authors express their gratitude to Springer, New York, for its highly professional assistance and above all we want to thank our editor, Mrs. Elizabeth Loew, for her strong moral support, patience and kind understanding.
Krakow, Poland Leszek Gasinski Athens, Greece Nikolaos S. Papageorgiou
Contents
1.1.2 Sequences and Complete Metric Spaces . . . . 4
1.1.3 Topology of Metric Spaces . . . . . . . . . . . . 4
1.1.4 Baire Theorem . . . . . . . . . . . . . . . . . . 8
1.1.6 Completion of Metric Spaces: Equivalence of Metrics . . . . . . . . . . . . . . . . . . . . . 14
1.1.7 Pointwise and Uniform Convergence of Maps . 16
1.1.8 Compact Metric Spaces . . . . . . . . . . . . . 17
1.1.9 Connectedness . . . . . . . . . . . . . . . . . . 22
1.1.11 Products of Metric Spaces . . . . . . . . . . . . 28
1.1.12 Auxiliary Notions . . . . . . . . . . . . . . . . 31
2.1.3 Nets . . . . . . . . . . . . . . . . . . . . . . . . 199
2.1.4 Continuous and Semicontinuous Functions . . . 201
2.1.5 Open and Closed Maps: Homeomorphisms . . 208 2.1.6 Weak (or Initial) and Strong (or Final)
Topologies . . . . . . . . . . . . . . . . . . . . . 209
vii
2.1.12 Michael Selection Theorem . . . . . . . . . . . 229
2.1.13 The Space C(X;Y ) . . . . . . . . . . . . . . . 230
2.1.14 Elements of Algebraic Topology I: Homotopy . 233
2.1.15 Elements of Algebraic Topology II: Homology . 245
2.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . 265
2.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 293
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 403
3.1.3 The Lebesgue Measure . . . . . . . . . . . . . . 412
3.1.4 Atoms and Nonatomic Measures . . . . . . . . 414
3.1.5 Product Measures . . . . . . . . . . . . . . . . 415
3.1.7 Measurable Functions . . . . . . . . . . . . . . 418
3.1.9 Convergence Theorems . . . . . . . . . . . . . . 430
3.1.13 Signed Measures . . . . . . . . . . . . . . . . . 446
3.1.15 Maximal Function and Lyapunov Convexity Theorem . . . . . . . . . . . . . . . . . . . . . 450
3.1.16 Conditional Expectation and Martingales . . . 451
3.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . 460
3.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 497
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 633
Contents ix
4.1.5 Polish, Souslin and Borel Spaces . . . . . . . . 640
4.1.6 Measurable Multifunctions: Selection Theorems 643
4.1.7 Projection Theorems . . . . . . . . . . . . . . . 647
4.1.9 Sequences of Measures: Weak Convergence in Lp(Ω) . . . . . . . . . . . . . . . . . . . . . . 649
4.1.10 Covering Theorems . . . . . . . . . . . . . . . . 652
4.1.12 Bounded Variation and Absolutely Continuous Functions . . . . . . . . . . . . . . . . . . . . . 658
4.1.13 Hausdorff Measures: Change of Variables . . . 672
4.1.14 Caratheodory Functions . . . . . . . . . . . . . 675
5.1.3 The Hahn–Banach Theorem . . . . . . . . . . . 840
5.1.4 Adjoint Operators and Annihilators . . . . . . 842
5.1.5 The Three Basic Theorems of Linear Functional Analysis . . . . . . . . . . . . . . . . . . . . . . 843
5.1.6 The Weak Topology . . . . . . . . . . . . . . . 846
5.1.7 The Weak∗ Topology . . . . . . . . . . . . . . . 848
5.1.8 Reflexive Banach Spaces . . . . . . . . . . . . . 850
5.1.9 Separable Banach Spaces . . . . . . . . . . . . 851
5.1.10 Uniformly Convex Spaces . . . . . . . . . . . . 853
5.1.11 Hilbert Spaces . . . . . . . . . . . . . . . . . . 854
5.1.13 Extremal Structure of Sets . . . . . . . . . . . 863
5.1.14 Compact Operators . . . . . . . . . . . . . . . 865
5.1.15 Spectral Theory . . . . . . . . . . . . . . . . . 867
5.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . 876 5.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 903
Bibliography . . . . . . . . . . . . . . . . . . . . . . . 1003
1.1.1 Basic Definitions and Notation
Definition 1.1 A metric space is a pair (X, dX ) of a set X and a function dX : X × X −→ R, which has the following properties: (a) d
X (x, y) 0 for all x, y ∈ X and d
X (x, y) = 0 if and only if
x = y; (b) d
X (x, y) = d
(c) d X (x, y) d
X (x, u) + d
X (u, y) for all x, y, u ∈ X (triangle in-
equality).
The function dX (·, ·) is called a metric or distance.
Remark 1.2 If (a) in the above definition is replaced by a weaker requirement: (a)′ dX (x, y) 0 for all x, y ∈ X and if x = y, then dX (x, y) = 0, then dX is said to be a pseudometric or ecart or semimetric and (X, d
X ) is a pseudometric space (or semimetric space).
Example 1.3
) }N k=1
X =
N∏
L. Gasinski and N.S. Papageorgiou, Exercises in Analysis: Part 1, Problem Books in Mathematics, DOI 10.1007/978-3-319-06176-4 1, © Springer International Publishing Switzerland 2014
1
and
(xk, yk) : 1 k N ) ∀ x, y ∈ X,
where 1 p < +∞, x = (xk) N k=1, y = (yk)
N k=1 ∈ X. Then (X, dp)
(1 p < +∞) and (X, d∞) are metric spaces. In particular, if Xk = R and
d R (x, y) = |x− y| ∀ x, y ∈ R,
then d R is a metric on R. For N 1, d2 is the Euclidean metric on
R N .
(b) Suppose now that we have an infinite family { (Xk, dXk
) } k1
sup { d Xk
We set X =
2k d Xk
(xk, yk) ∀ x = {xk}k1 , y = {yk}k1 ∈ X.
Then (X, d X ) is a metric space.
(c) Let X be an arbitrary nonempty set and let
dd X (x, y) =
∀ x, y ∈ X.
Then (X, dd X ) is a metric space and dd
X is called the discrete metric
on X.
(d) Let X = C[a, b] (the space of continuous functions on [a, b]). We set
d1 X (f, g) =
X (f, g) = max
1.1. Introduction 3
X ) are metric spaces. However, as we will
see later (see Remark 1.82) these metric spaces are fundamentally dif- ferent. We call d1
X the L1-metric and d∞
X the uniform metric or
supremum metric on X = C[a, b].
(e) Let X = R ∪ {±∞} (extended real line) and set
d X (x, y) =
tan−1 x− tan−1 y ∀ x, y ∈ X
(recall that tan−1(±∞) = ±π 2 and tan−1 is injection). Then (X, d
X )
X ) is a metric space and
d X (x, y) =
d X (x, y)
∀ x, y ∈ X,
Remark 1.5 Note that
dX (x, y) < 1 ∀ x, y ∈ X. Moreover, using this proposition, we can see that the space X of all real sequences equipped with the distance
d X (x, y) =
Definition 1.6 Let (X, d
X ) be a metric space.
(a) The open ball centred at x0 ∈ X of radius r > 0 is defined by
Br(x0) = { x ∈ X : d
X (x, x0) < r
} .
(b) A set C ⊆ X is said to be bounded if it is contained in some open ball. The set C is unbounded if this is not the case.
4 Chapter 1. Metric Spaces
(c) The diameter of a set C ⊆ X is given by
diamC = sup { d X (x, y) : x, y ∈ C}
(if C = ∅, then diamC = 0). (d) For any x ∈ X and A,B ⊆ X, we define:
dist(x,A) = inf { d X (x, a) : a ∈ A}
and
dist(A,B) = inf { d X (a, b) : a ∈ A, b ∈ B}.
1.1.2 Sequences and Complete Metric Spaces
Definition 1.7 Let (X, d
X ) be a metric space and let {xn}n1 ⊆ X be a sequence.
(a) We say that the sequence {xn}n1 ⊆ X converges to x ∈ X if and only if
dX (xn, x) −→ 0 as n→ +∞, i.e., for any r > 0, we can find an integer n0 = n0(r) 1 such that
xn ∈ Br(x) ∀ n n0.
The point x ∈ X is said to be the limit of the convergent sequence {xn}n1. (b) Let (X, d
X ) be a metric space. A sequence {xn}n1 ⊆ X is said to
be a Cauchy sequence if for any given ε > 0, we can find an integer n0 = n0(ε) 1 such that
d X (xn, xk) ε ∀ n, k n0.
(c) A metric space (X, d X ) is said to be complete if every Cauchy
sequence in X is convergent in X.
1.1.3 Topology of Metric Spaces
To be able to proceed further with the study of metric spaces, we need to introduce some topological material associated with them. We will return to these concepts in a more general setting in Chap. 2.
1.1. Introduction 5
Definition 1.8 Let (X, d
X ) be a metric space.
(a) A set U ⊆ X is said to be open if for every x ∈ U we can find r = r(x) > 0 such that Br(x) ⊆ U . (b) A set C ⊆ X is said to be closed if the set X \ C is open. (c) The family of open sets U ⊆ X is called the topology determined by the metric d
X (metric topology).
(d) A set A ⊆ X is called a neighbourhood of a point x ∈ X if there exists r > 0 such that Br(x) ⊆ A.
Proposition 1.9 If (X, dX ) is a metric space, then (a) ∅ and X are open sets;
(b) if {Ui}i∈I is any family of open sets in X, then
i∈I Ui is an open
set too;
(c) if {Uk}Nk=1 is any finite family of open sets in X, then
N
Using de Morgan laws and Definition 1.8(b), we also have
Proposition 1.10 If (X, d
X ) is a metric space, then
(a) ∅ and X are closed sets;
(b) if {Ci}i∈I is any family of closed sets in X, then
i∈I Ci is a closed
set too;
(c) if {Ck}Nk=1 is any finite family of closed sets in X, then N
k=1
Ck is
We can characterize sets in terms of convergent sequences.
Proposition 1.11 If (X, dX ) is a metric space, then C ⊆ X is closed if and only if every convergent sequence {xn}n1 ⊆ C has limit in C.
6 Chapter 1. Metric Spaces
Definition 1.12 Let (X, d
X ) be a metric space and E ⊆ X. The subspace (metric)
topology on E induced by the metric d X
is the family
} .
X ) be a metric space and let E ⊆ X.
(a) We say that x ∈ E is an interior point of E if there exists r > 0 such that Br(x) ⊆ E. The set of all interior points of E is called the interior of E and is denoted by intE. (b) We say that x ∈ E is a limit (or cluster or accumulation) point of E if Br(x) ∩
( E \ {x}) = ∅ for every r > 0. The union of E
with the set of all its accumulation points is called the closure of E and is denoted by E. (c) The boundary of E is defined by ∂E = E ∩ Ec. (d) We say that x ∈ E is an isolated point of E, if there is an r > 0 such that Br(x)∩
( E \ {x}) = ∅. Hence {x} is a relatively open subset
of E. (e) We say that E ⊆ X is a perfect set if every point of E is an accumulation point.
Theorem 1.14 Let (X,…
Series editor:
Peter Winkler Department of Mathematics Dartmouth College Hanover, NH 03755 USA
For further volumes: http://www.springer.com/series/714
Exercises in Analysis
and Computer Science Jagiellonian University Krakow, Poland
Nikolaos S. Papageorgiou Department of Mathematics National Technical University Athens, Greece
ISSN 0941-3502 ISBN 978-3-319-06175-7 ISBN 978-3-319-06176-4 (eBook) DOI 10.1007/978-3-319-06176-4 Springer Cham Heidelberg New York Dordrecht London
Library of Congress Control Number: 2014940530
Mathematics Subject Classification (2010): 00A07, 46-XX, 30LXX, 28-XX, 60G46
© Springer International Publishing Switzerland 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publica- tion does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of pub- lication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.
Printed on acid-free paper
The aim of this book is to review the theory of some basic topics in Analysis and accompany the theory with problems and their solutions. With the problems the reader can test his/her understanding of the theory and also discover extensions of the theory and additional results which are not so standard in the literature. The topics covered span a more or less standard advanced undergraduate and graduate curricu- lum in Analysis. More precisely, we focus on the following subjects:
1. Metric Spaces
2. Topological Spaces (here there is also some introductory material on Algebraic Topology)
3. Measure, Integration and Martingales (including Lp-spaces)
4. Measure and Topology (covering issues concerning the interplay between measure theory and topology)
5. Functional Analysis (with emphasis on basic Banach space theory).
Each one of the above five subjects corresponds to a different chap- ter. In the first part of each chapter, we present the basic theory, with all the main definitions and results. We also include comments and remarks expanding on the concepts and results, but no proofs. This review material will help the reader refresh his/her knowledge of the theory before tackling the problems. In each chapter the theory is fol- lowed by problems and their detailed solution. In each chapter, there are at least 170 problems, marked with , or according to their difficulty. Some of those problems complement the theory, while the rest check the reader’s understanding of the theory. We strongly encourage the reader to put some substantial personal effort in trying to solve the problem, before looking at the solution. Otherwise, they will get no benefit from reading the book. On the other hand, a serious
v
vi Preface
effort to solve the problem themselves and subsequently compare their solution with the one provided (or checking to see where they failed to come up with the right arguments to produce a solution) will help them to achieve a solid understanding of the theory.
It is not easy to provide the origin of each of the problems and of their solutions. They can be traced as problems included in the books mentioned in the literature (where they are stated without proofs), or they can be found in the problem books listed in the References, or they are standard exercises in the public domain, or they were accumulated through the years from teaching undergraduate and graduate courses on these or closely related subjects. In any case, the books mentioned in the References can be a valuable source for additional theoretical material and more problems. Our book is only the starting point (helpful we hope).
The authors express their gratitude to Springer, New York, for its highly professional assistance and above all we want to thank our editor, Mrs. Elizabeth Loew, for her strong moral support, patience and kind understanding.
Krakow, Poland Leszek Gasinski Athens, Greece Nikolaos S. Papageorgiou
Contents
1.1.2 Sequences and Complete Metric Spaces . . . . 4
1.1.3 Topology of Metric Spaces . . . . . . . . . . . . 4
1.1.4 Baire Theorem . . . . . . . . . . . . . . . . . . 8
1.1.6 Completion of Metric Spaces: Equivalence of Metrics . . . . . . . . . . . . . . . . . . . . . 14
1.1.7 Pointwise and Uniform Convergence of Maps . 16
1.1.8 Compact Metric Spaces . . . . . . . . . . . . . 17
1.1.9 Connectedness . . . . . . . . . . . . . . . . . . 22
1.1.11 Products of Metric Spaces . . . . . . . . . . . . 28
1.1.12 Auxiliary Notions . . . . . . . . . . . . . . . . 31
2.1.3 Nets . . . . . . . . . . . . . . . . . . . . . . . . 199
2.1.4 Continuous and Semicontinuous Functions . . . 201
2.1.5 Open and Closed Maps: Homeomorphisms . . 208 2.1.6 Weak (or Initial) and Strong (or Final)
Topologies . . . . . . . . . . . . . . . . . . . . . 209
vii
2.1.12 Michael Selection Theorem . . . . . . . . . . . 229
2.1.13 The Space C(X;Y ) . . . . . . . . . . . . . . . 230
2.1.14 Elements of Algebraic Topology I: Homotopy . 233
2.1.15 Elements of Algebraic Topology II: Homology . 245
2.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . 265
2.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 293
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 403
3.1.3 The Lebesgue Measure . . . . . . . . . . . . . . 412
3.1.4 Atoms and Nonatomic Measures . . . . . . . . 414
3.1.5 Product Measures . . . . . . . . . . . . . . . . 415
3.1.7 Measurable Functions . . . . . . . . . . . . . . 418
3.1.9 Convergence Theorems . . . . . . . . . . . . . . 430
3.1.13 Signed Measures . . . . . . . . . . . . . . . . . 446
3.1.15 Maximal Function and Lyapunov Convexity Theorem . . . . . . . . . . . . . . . . . . . . . 450
3.1.16 Conditional Expectation and Martingales . . . 451
3.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . 460
3.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 497
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 633
Contents ix
4.1.5 Polish, Souslin and Borel Spaces . . . . . . . . 640
4.1.6 Measurable Multifunctions: Selection Theorems 643
4.1.7 Projection Theorems . . . . . . . . . . . . . . . 647
4.1.9 Sequences of Measures: Weak Convergence in Lp(Ω) . . . . . . . . . . . . . . . . . . . . . . 649
4.1.10 Covering Theorems . . . . . . . . . . . . . . . . 652
4.1.12 Bounded Variation and Absolutely Continuous Functions . . . . . . . . . . . . . . . . . . . . . 658
4.1.13 Hausdorff Measures: Change of Variables . . . 672
4.1.14 Caratheodory Functions . . . . . . . . . . . . . 675
5.1.3 The Hahn–Banach Theorem . . . . . . . . . . . 840
5.1.4 Adjoint Operators and Annihilators . . . . . . 842
5.1.5 The Three Basic Theorems of Linear Functional Analysis . . . . . . . . . . . . . . . . . . . . . . 843
5.1.6 The Weak Topology . . . . . . . . . . . . . . . 846
5.1.7 The Weak∗ Topology . . . . . . . . . . . . . . . 848
5.1.8 Reflexive Banach Spaces . . . . . . . . . . . . . 850
5.1.9 Separable Banach Spaces . . . . . . . . . . . . 851
5.1.10 Uniformly Convex Spaces . . . . . . . . . . . . 853
5.1.11 Hilbert Spaces . . . . . . . . . . . . . . . . . . 854
5.1.13 Extremal Structure of Sets . . . . . . . . . . . 863
5.1.14 Compact Operators . . . . . . . . . . . . . . . 865
5.1.15 Spectral Theory . . . . . . . . . . . . . . . . . 867
5.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . 876 5.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 903
Bibliography . . . . . . . . . . . . . . . . . . . . . . . 1003
1.1.1 Basic Definitions and Notation
Definition 1.1 A metric space is a pair (X, dX ) of a set X and a function dX : X × X −→ R, which has the following properties: (a) d
X (x, y) 0 for all x, y ∈ X and d
X (x, y) = 0 if and only if
x = y; (b) d
X (x, y) = d
(c) d X (x, y) d
X (x, u) + d
X (u, y) for all x, y, u ∈ X (triangle in-
equality).
The function dX (·, ·) is called a metric or distance.
Remark 1.2 If (a) in the above definition is replaced by a weaker requirement: (a)′ dX (x, y) 0 for all x, y ∈ X and if x = y, then dX (x, y) = 0, then dX is said to be a pseudometric or ecart or semimetric and (X, d
X ) is a pseudometric space (or semimetric space).
Example 1.3
) }N k=1
X =
N∏
L. Gasinski and N.S. Papageorgiou, Exercises in Analysis: Part 1, Problem Books in Mathematics, DOI 10.1007/978-3-319-06176-4 1, © Springer International Publishing Switzerland 2014
1
and
(xk, yk) : 1 k N ) ∀ x, y ∈ X,
where 1 p < +∞, x = (xk) N k=1, y = (yk)
N k=1 ∈ X. Then (X, dp)
(1 p < +∞) and (X, d∞) are metric spaces. In particular, if Xk = R and
d R (x, y) = |x− y| ∀ x, y ∈ R,
then d R is a metric on R. For N 1, d2 is the Euclidean metric on
R N .
(b) Suppose now that we have an infinite family { (Xk, dXk
) } k1
sup { d Xk
We set X =
2k d Xk
(xk, yk) ∀ x = {xk}k1 , y = {yk}k1 ∈ X.
Then (X, d X ) is a metric space.
(c) Let X be an arbitrary nonempty set and let
dd X (x, y) =
∀ x, y ∈ X.
Then (X, dd X ) is a metric space and dd
X is called the discrete metric
on X.
(d) Let X = C[a, b] (the space of continuous functions on [a, b]). We set
d1 X (f, g) =
X (f, g) = max
1.1. Introduction 3
X ) are metric spaces. However, as we will
see later (see Remark 1.82) these metric spaces are fundamentally dif- ferent. We call d1
X the L1-metric and d∞
X the uniform metric or
supremum metric on X = C[a, b].
(e) Let X = R ∪ {±∞} (extended real line) and set
d X (x, y) =
tan−1 x− tan−1 y ∀ x, y ∈ X
(recall that tan−1(±∞) = ±π 2 and tan−1 is injection). Then (X, d
X )
X ) is a metric space and
d X (x, y) =
d X (x, y)
∀ x, y ∈ X,
Remark 1.5 Note that
dX (x, y) < 1 ∀ x, y ∈ X. Moreover, using this proposition, we can see that the space X of all real sequences equipped with the distance
d X (x, y) =
Definition 1.6 Let (X, d
X ) be a metric space.
(a) The open ball centred at x0 ∈ X of radius r > 0 is defined by
Br(x0) = { x ∈ X : d
X (x, x0) < r
} .
(b) A set C ⊆ X is said to be bounded if it is contained in some open ball. The set C is unbounded if this is not the case.
4 Chapter 1. Metric Spaces
(c) The diameter of a set C ⊆ X is given by
diamC = sup { d X (x, y) : x, y ∈ C}
(if C = ∅, then diamC = 0). (d) For any x ∈ X and A,B ⊆ X, we define:
dist(x,A) = inf { d X (x, a) : a ∈ A}
and
dist(A,B) = inf { d X (a, b) : a ∈ A, b ∈ B}.
1.1.2 Sequences and Complete Metric Spaces
Definition 1.7 Let (X, d
X ) be a metric space and let {xn}n1 ⊆ X be a sequence.
(a) We say that the sequence {xn}n1 ⊆ X converges to x ∈ X if and only if
dX (xn, x) −→ 0 as n→ +∞, i.e., for any r > 0, we can find an integer n0 = n0(r) 1 such that
xn ∈ Br(x) ∀ n n0.
The point x ∈ X is said to be the limit of the convergent sequence {xn}n1. (b) Let (X, d
X ) be a metric space. A sequence {xn}n1 ⊆ X is said to
be a Cauchy sequence if for any given ε > 0, we can find an integer n0 = n0(ε) 1 such that
d X (xn, xk) ε ∀ n, k n0.
(c) A metric space (X, d X ) is said to be complete if every Cauchy
sequence in X is convergent in X.
1.1.3 Topology of Metric Spaces
To be able to proceed further with the study of metric spaces, we need to introduce some topological material associated with them. We will return to these concepts in a more general setting in Chap. 2.
1.1. Introduction 5
Definition 1.8 Let (X, d
X ) be a metric space.
(a) A set U ⊆ X is said to be open if for every x ∈ U we can find r = r(x) > 0 such that Br(x) ⊆ U . (b) A set C ⊆ X is said to be closed if the set X \ C is open. (c) The family of open sets U ⊆ X is called the topology determined by the metric d
X (metric topology).
(d) A set A ⊆ X is called a neighbourhood of a point x ∈ X if there exists r > 0 such that Br(x) ⊆ A.
Proposition 1.9 If (X, dX ) is a metric space, then (a) ∅ and X are open sets;
(b) if {Ui}i∈I is any family of open sets in X, then
i∈I Ui is an open
set too;
(c) if {Uk}Nk=1 is any finite family of open sets in X, then
N
Using de Morgan laws and Definition 1.8(b), we also have
Proposition 1.10 If (X, d
X ) is a metric space, then
(a) ∅ and X are closed sets;
(b) if {Ci}i∈I is any family of closed sets in X, then
i∈I Ci is a closed
set too;
(c) if {Ck}Nk=1 is any finite family of closed sets in X, then N
k=1
Ck is
We can characterize sets in terms of convergent sequences.
Proposition 1.11 If (X, dX ) is a metric space, then C ⊆ X is closed if and only if every convergent sequence {xn}n1 ⊆ C has limit in C.
6 Chapter 1. Metric Spaces
Definition 1.12 Let (X, d
X ) be a metric space and E ⊆ X. The subspace (metric)
topology on E induced by the metric d X
is the family
} .
X ) be a metric space and let E ⊆ X.
(a) We say that x ∈ E is an interior point of E if there exists r > 0 such that Br(x) ⊆ E. The set of all interior points of E is called the interior of E and is denoted by intE. (b) We say that x ∈ E is a limit (or cluster or accumulation) point of E if Br(x) ∩
( E \ {x}) = ∅ for every r > 0. The union of E
with the set of all its accumulation points is called the closure of E and is denoted by E. (c) The boundary of E is defined by ∂E = E ∩ Ec. (d) We say that x ∈ E is an isolated point of E, if there is an r > 0 such that Br(x)∩
( E \ {x}) = ∅. Hence {x} is a relatively open subset
of E. (e) We say that E ⊆ X is a perfect set if every point of E is an accumulation point.
Theorem 1.14 Let (X,…