Lipschitz Functions

604 Pages·2019·26.063 MB·English

by  Cobzaş, Ş.;Miculescu, R.;Nicolae, A.

Lecture Notes in Mathematics 2241
Editors-in-Chief:
Jean-Michel Morel, Cachan
BernardTeissier, Paris
Advisory Editors:
Michel Brion, Grenoble
Camillo De Lellis, Princeton
Alessio Figalli, Zurich
Davar Khoshnevisan, Salt Lake City
Ioannis Kontoyiannis, Cambridge
GaborLugosi, Barcelona
Mark Podolskij, Aarhus
Sylvia Serfaty, New York
Anna Wienhard, Heidelberg
Moreinformation aboutthis series at http://www.springer.com/series/304
Stefan Cobzas ¢ Radu Miculescu * Adriana Nicolae
Lipschitz Functions
go) Springer
Stefan Cobzag Radu Miculescu
Faculty ofMathematics and Computer Faculty ofMathematics and Computer
Science Science
Babes-Bolyai University Transilvania University ofBrasov
Cluj-Napoca, Romania Brasov, Romania
Adriana Nicolae
Faculty ofMathematics and Computer
Science
Babes-Bolyai University
Cluj-Napoca, Romania
ISSN 0075-8434 ISSN 1617-9692 (electronic)
Lecture Notes in Mathematics
ISBN 978-3-030-16488-1 ISHN 978-3-030-16489-8 (eBook)
https://doi.org/10.1007/978-3-030-16489-8
Library ofCongress Control Number: 2019936365
Mathematics Subject Classification (2010): Primary: 46-02, 26A16, 30L05, 46A22, 46B20, 46B22,
46B80, 46B85; Secondary: 46E15, 46E40, 47H09, 47B33, 47B07, 53C22, 54C20, 54C65
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Preface
The aim of this book is to give an essentially self-contained account of the main
classical results in the theoryofLipschitzfunctions. In fact, this projectoriginatedas
an outgrowthofamastercoursetaughtby the second-namedauthorat theUniversity
ofBucharest. In time, we developedit and addressed additional relevant topics and
recent research trends concerningthis class offunctions.
The prerequisites are basic results in real analysis, functional analysis, measure
theory (including vector measures), and topology, which, for readers’ convenience,
are surveyedin the first chapter ofthe book, Chap. 1, “Prerequisites”.
Lipschitz functions form a class of functions which appear not only in many
branches of mathematics, as the theory of ordinary differential equations, partial
differential equations, measure theory, nonlinear functional analysis, topology,
metric geometry, and fractal theory, but also in computer science, as in image
processingor in the study ofInternet search engines.
Taking into account the classical theorem of H. Rademacher whichstates that a
Lipschitz function f : 82 — R”, where $2 is an open subset ofR”,is differentiable
outside of a Lebesgue null subset of £2, the condition of being Lipschitz could be
viewed as a weakened version of differentiability, and therefore, these functions are
a good substitute for smooth functions in the framework ofmetnic spaces.
Chapter 2 contains some basic results concerning Lipschitz and locally Lips-
chitz functions—algebraic operations, sequences of Lipschitz functions, Lipschitz
properties for differentiable functions (including a characterization in terms ofDini
derivatives), or gluing Lipschitz functions together. The existence of Lipschitz and
locally Lipschitz partitions ofunity with applications to sandwich-type theorems,to
Lipschitz selections ofset-valued mappings,and to the Lipschitz separability ofthe
Banach space C(T) is also proved.
Chapter 3 starts with a detailed discussion on Lipschitz properties of convex
functions, including vector functions. In the vector case, meaning convex functions
defined on a locally convex space with values in a locally convex space ordered
by a cone, we emphasize the key role played by the normality of the cone. Equi-
Lipschitz properties of families of convex functions and Lipschitz properties of
convex functions defined on metric linear spaces are discussed as well.
vi Preface
Other considered topics involve the existence of an equivalent metric making
a given continuous function Lipschitz and metric spaces where every continuous
function is Lipschitz. An old result of Fichtenholz (from 1922) on the relation
between absolutely continuous and Lipschitz functions is included. The chapter
ends with a discussion on the differentiability properties of Lipschitz functions—
Rademacher-typetheorems—infinite and in infinite dimension.
Thepossibility to extend a Lipschitz function from a subset of a metric space to
the whole space with thepreservationofthe Lipschitz constant(aHahn-Banachtype
result for Lipschitz functions) is studied in Chap. 4. This chapter contains several
results on the existence of norm-preserving extensions of Lipschitz functions—
Kirszbraun, McShane, Valentine, and Flett. A discussion on the corresponding
property for semi-Lipschitz functions defined on quasi-metric spaces and for
Lipschitz functions with values in a quasi-normed spaceis included as well.
Chapter 5 is concerned with Lipschitz functions on geodesic metric spaces,
which are a natural generalization of Riemannian manifolds and provide a suitable
setting for the study of problems from various areas ofmathematics with important
applications. We review in this chapter some selected properties of Lipschitz
mappings in geodesic metric spaces, focusingmainly on certain extension theorems
which generalize corresponding ones from linear contexts.
After introducing some definitions and results from thetheory ofgeodesic metric
spaces with an emphasis on the notion of curvature, we discuss Lipschitz extension
results of Kirszbraun and McShanetype in Alexandrov spaces with lower or upper
curvature bounds consideredglobally. Even ifthe existence ofa Lipschitz extension
is guaranteed by an extension result, in general this extension is not unique. Here,
we address the parameter dependenceof extensions of Lipschitz mappings from
the point of view of continuity (with respect to the supremum distance). This
chapter additionally includes two counterparts of the Dugundji extension theorem
for continuous orLipschitz mappings with values in nonpositively curved spaces in
the sense ofBusemann.
Chapter 6 deals with the possibility to approximate various classes of functions
(e.g., uniformly continuous) by Lipschitz functions, based on Lipschitz partitions
of unity or on some extension results for Lipschitz functions. A result due to Baire
on the approximation of semicontinuous functions by continuous ones, based on
McShane’s extension method,is also included. This chapter also contains a study
of the homotopy of Lipschitz functions (two homotopic Lipschitz functions are
Lipschitz homotopic) and an introduction to Lipschitz manifolds.
The main results ofChap. 7 are Aharoni’s result (from 1974) on the bi-Lipschitz
embeddability of separable metric spaces in the Banach space co anda result of
Vaisala (from 1992) on the characterization of the completeness of a normed space
X by the non-existence ofbi-Lipschitz surjections of X onto X \ {0}. Other related
results are discussed in the final section of the chapter. The chapter offers only
a glimpse of this very active area of research, the topic being treated at large in
the books by Benyamini and Lindenstrauss [75] and in the two-volumetreatise by
Brudnyi and Brudnyi[126, 127].
Preface Vii
The validity of an extension result ofHahn-Banach type for Lipschitz functions
makes the space of Lipschitz functions on a Banach space X a good substitute for
the linear dual X*. This idea, combined with the method of Lipschitz free Banach
spaces, madepossible the extension ofmany results in functional analysis from the
linear case to the Lipschitz one, a topic treated in Chap.8.
We introduceseveral Banach spacesofLipschitz functions (Lipschitz functions
vanishing at a fixed point, bounded Lipschitz functions,little Lipschitz functions)
on a metric space and present some of their properties. A detailed study of free
Lipschitz spaces is carried out, including several ways to introduce them and
corresponding duality results. The study of the Kantorovich-Rubinstein and Hanin
normsis tightly connected with Lipschitz spaces, mainly via the weak convergence
of probability measures, a topic treated here in detail. The case of functions with
values in a Hilbert space is considered as well, the key tool for the treatment ofthis
case being a sesquilinear integral for Hilbert space-valued functions.
Compactness and weak compactnessproperties ofLipschitz operators on Banach
spaces and of composition operators on spaces of Lipschitz functions are also
studied, emphasizing the key role played by Lipschitz free Banach spaces. Another
themepresented here is the Bishop-Phelps property for Lipschitz functions, mean-
ing density results for Lipschitz functionsthat attain their Lipschitz norm.
Applications to best approximation in metric spaces and in metric linear spaces
X are given in the last section of this chapter, where it is shown how results from
the linear case can be transposedto this situation by using as a dual space the space
ofLipschitz functions defined on X.
There are two other books devotedto Lipschitzfunctions and spaces ofLipschitz
functions—the book by Weaver[675] and that by Miculescu andMortici [482]. We
tried to keep the overlapping with these booksat an inevitable minimum, making
this book complementary to them. An important topic missing from this book is
that offixed points for Lipschitz mappings, but whichis well treated in many books
devoted to fixed point theory, as, for instance, in [570].
The bibliography (almost 700 items) contains references to the sources of the
results included in the book as well as to further results mentioned in the final
sections of each chapter, in order to help the potential reader to get acquainted
with the current status of the subject and to find his own line of investigation. In
spite of its wealth, the bibliography is far from bemg complete, but we strived to
be as accurate as possible in attributing a result to the nght person. We apologize
in advance for any inadvertence. An exception is the chapter “Prerequisites”, where
references to some booksrather than to original papers are given. At the beginning
of each section in this chapter, the sources on which the presentation is based are
indicated.
The bookis accessible to graduate students (some parts even to undergraduates),
but it also contains recent results of interest to researchers in various domains—
metric geometry, mathematical analysis, and functional analysis. The book (or
parts of it) is also suitable as a support for graduate or advanced undergraduate
courses. We hopethat it will be ofinterest to everyone whose domain ofinterestis
mathematical analysis and its applications.
Viii Preface
The authors express their thanks to the reviewers for the careful reading of
the manuscript and for the remarks and suggestions that led to a substantial
improvementof the presentation, in both style and contents. Our warmest thanks
also go to the Springer Editors for the professional cooperation, especially to Ute
McCrory whose support and encouragements helped usreachthisfinal stage.
Cluj-Napoca, Romania Stefan Cobzas
Brasov, Romania Radu Miculescu
Cluj-Napoca, Romania AdrianaNicolae
January 30, 2019
Contents
1 Prerequisites ........ 0... cece cence eee eee e eee eee e nese enon eeeeeeees 1
1.1 Ordered Sets ........ 0... cece cece cece eenneeneenneeeeeeeseseeeeeeeees 1
1.1.1 Preorder and Order .........ccc cccseccc cceeee nese eeeneeeeeeeees 1
1.1.2 OrderedVector Spaces ............ ccs cccceeeeceseeeeeeeeeeees 2
1.1.3 Convex Sets and Convex Functions ...........ccceeeeeeeeeees 6
1.1.4 The Minkowski Functional, Norms and Seminorms........ 8
1.1.5 Limit Inferior and Limit Superior of Sequences
ofReal Numbers............. 0... cece cece cc eeeeee eeeneeeeeeees 9
1.2 Topological Spaces............cccccceeccccee ence eee e eee e eee eeeeeeneneee 10
1.2.1. The Notion ofTopological Space..............ccccceeeeeeeeee 10
1.2.2 Separation AXIOMS .......... 0. ccc cece eee e cece e eee eeeeeeeeeeees 12
1.2.3 COMPACINESS ........ cece cece cece sees eeen ween eenenneennees 13
1.2.4 Continuous FUNCTIONS ..........cccc eee e cece cece eeeeneeeeeneees 13
1.2.5 Semicontinuous FUNCTIONS ...........cc cece cece eeeeeeeeeeeeees 15
1.2.6 Sequences and Nets in Topological Spaces ...............64: 16
1.2.7 Products ofTopological Spaces. Tihonov’s Theorem ....... 17
1.3 Metric Spaces 2.0.0... cece cece cece cere e eee ee eee et esse eee eeenees 19
1.3.1 The Notion ofMetric Space ............cc cece cece ene e ee eeaees 19
1.3.2. Uniformly Continuous, Lipschitz and Holder Functions..... 21
1.3.3. The Distance FUMCtion ......... ccc cece cc cececeeeeneeeeeeeeeees 24
1.3.4 The Pompeiu-HausdorffMetric ..............ccceeee eeee ee ees 25
1.3.5. Characterizations ofContinuity in the Metric Case ......... 27
1.3.6 Completeness and Baire Category .................... 2.2. e eee 29
1.3.7. Compactness in Metric Spaces ................cceeeeee cece ees 30
1.3.8 Equivalent Metrics .......... 2.0...e ccc cceece cece eeeeeees 31
1.3.9 Ultrametric Spaces ..............ccc ccc ccce cece ceeene eee eeeanes 33
1.3.10 Paracompact Spaces ............. 00...cececence seen eeeeeeeeees 34
1.3.11 Partitions ofUnity ..........ccccccce cece eee e eee eeeeeeeeeeennees 36
1.3.12 Sandwich and Approximation Results
for Semicontinuous FUNCTIONS ...........cccceeeeeee nese eeeees 36
1x