Graduate Texts in Mathematics 183

Springer-Verlag New York, Inc.

October, 1998

**Acknowledgment: **I wish to
express my gratitude to Allen Bryant, who worked through the initial part of
Chapter 2 while a graduate student at Eastern Illinois University and caught
several errors that were corrected before this book saw the light of day.
Allen's name was unintentionally omitted from the list of those whom I thanked
in the preface of the text, which will be corrected in future printings. Thanks
again, Allen, for your help.

Portions of the following are excerpts from the preface of this book, and are © 1998 Springer-Verlag New York, Inc.

The purpose of this book is to serve as a text for a graduate
course in functional analysis emphasizing Banach space theory.
Its intended audience is graduate students who have had the standard
courses in analysis and measure theory up to and including elementary
properties of the *L*_*p* spaces, but who may not yet
have seen any of the basic results from a first course in functional
analysis, such as the uniform boundedness principle and the various
forms of the Hahn-Banach theorem. A detailed description of the
prerequisites for reading this text
is given below.

Though many of the classical results of functional analysis are presented in this text, these results are applied for the most part to normed spaces in general and Banach spaces in particular, allowing a much more extensive development of that theory while placing correspondingly less emphasis on other topics that might appear in a traditional functional analysis text. For more information on the contents of the text, see the Table of Contents and section-by-section synopsis below.

The theory is developed for normed spaces over both the real and complex scalar fields. When a result holds for incomplete normed spaces as well as Banach spaces, the result is usually stated and proved in the more general form so that the reader will know where completeness is truly essential. However, results that can be extended from Banach spaces or arbitrary normed spaces to larger classes of topological vector spaces usually do not get the more general treatment unless the extension has a specific application to Banach space theory.

This book is sprinkled liberally with examples, both to show the theory at work and to illustrate why certain hypotheses in theorems are necessary. The book is also sprinkled liberally with historical notes and citations of original sources, with special attention given to mentioning dates within the body of the text so that the reader can get a feeling for the time frame within which the different parts of Banach space theory evolved.

Over 450 exercises are included, to provide the student with some practice in the use of the results developed in the text, to extend the theory presented in the text, and to present alternative developments of some of the main results. For example, in one of the exercises the reader is guided through Hahn's proof of the uniform boundedness principle, which is based on a gliding hump argument and does not use the Baire category theorem in any form.

Two asterisks preceding a section name indicates a section that is optional in the sense that no non-optional section depends on it.

**Preface****Basic Concepts**- Preliminaries
- Norms
- First Properties of Normed Spaces
- Linear Operators Between Normed Spaces
- Baire Category
- Three Fundamental Theorems
- Quotient Spaces
- Direct Sums
- The Hahn-Banach Extension Theorems
- Dual Spaces
- The Second Dual and Reflexivity
- Separability
- **Characterizations of Reflexivity

**The Weak and Weak* Topologies**- Topology and Nets
- Vector Topologies
- **Metrizable Vector Topologies
- Topologies Induced by Families of Functions
- The Weak Topology
- The Weak* Topology
- The Bounded Weak* Topology
- Weak Compactness
- **James's Weak Compactness Theorem
- Extreme Points
- **Support Points and Subreflexivity

**Linear Operators**- Adjoint Operators
- Projections and Complemented Subspaces
- Banach Algebras and Spectra
- Compact Operators
- Weakly Compact Operators

**Schauder Bases**- First Properties of Schauder Bases
- Unconditional Bases
- Equivalent Bases
- Bases and Duality
- **James's Space
*J*

**Rotundity and Smoothness**- Rotundity
- Uniform Rotundity
- Generalizations of Uniform Rotundity
- Smoothness
- Uniform Smoothness
- Generalizations of Uniform Smoothness

**APPENDIX A: Prerequisites****APPENDIX B: Metric Spaces****APPENDIX C: The Spaces \ell_***p*and \ell_*p*^*n*, 1 \le*p*\le \infty.**APPENDIX D: Ultranets****References****List of Symbols****Index**

Anyone who has studied the first third of Walter Rudin's *Real
and Complex Analysis,* which is to say the first six chapters
of that book, will be able to read this book through, cover-to-cover,
omitting nothing. Of course, this implies that the reader has
had the basic grounding in undergraduate mathematics necessary
to tackle Rudin's book, which should include a first course in
linear algebra. Though some knowledge of elementary topology beyond
the theory of metric spaces is assumed, the topology presented
near the beginning of Rudin's book is enough. In short, all of
this book is accessible to someone who has had a course in real
and complex analysis that includes the duality between the Lebesgue
spaces *L*_*p* and *L*_*q* when 1<*p*<\infty
and 1/*p*+1/*q*=1, as well as the Riesz representation
theorem for bounded linear functionals on *C*(*K*) where
*K* is a compact Hausdorff space, and who has not slighted
the usual prerequisites for such a course.

**Chapter 1** focuses on the metric theory of normed spaces.
The first three sections present fundamental definitions and examples,
as well as the most elementary properties of normed spaces such
as the continuity of their vector space operations. The fourth
section contains a short development of the most basic properties
of bounded linear operators between normed spaces, including properties
of normed space isomorphisms, which are then used to show that
every finite-dimensional normed space is a Banach space.

The Baire category theorem for nonempty complete metric spaces is the subject of Section 1.5. This section is, in a sense, optional, since none of the results outside of optional sections of this book depend directly on it, though some such results do depend on a weak form of the Baire category theorem that will be mentioned in the next paragraph. However, this section has not been marked optional, since a student far enough along in his or her mathematical career to be reading this book should become familiar with Baire category. This section is placed just before the section on the open mapping theorem, closed graph theorem, and uniform boundedness principle for the benefit of the instructor wishing to substitute traditional Baire category proofs of those results for the ones given here.

Since a course in functional analysis is not a prerequisite
for this book, the reader may not have seen the open mapping theorem,
closed graph theorem, and uniform boundedness principle for Banach
spaces. Section 1.6 is devoted to the development of those results.
All three are based on a very specific and easily proved form
of the Baire category theorem, presented in Section 1.3: *Every
closed, convex, absorbing subset of a Banach space includes a
neighborhood of the origin.*

In Section 1.7, the properties of quotient spaces formed from
normed spaces are examined and the first isomorphism theorem for
Banach spaces is proved: If *T* is a bounded linear operator
from a Banach space *X* onto a Banach space *Y*, then
*Y* and *X*/ker(*T*) are isomorphic as Banach spaces.
Following a section devoted to direct sums of normed spaces, Section
1.9 presents the vector space and normed space versions of the
Hahn-Banach extension theorem, along with their close relative,
Helly's theorem for bounded linear functionals. The same section
contains a development of Minkowski functionals and gives an example
of how they are used to prove versions of the Hahn-Banach separation
theorem. Section 1.10 introduces the dual space of a normed space,
and has the characterizations up to isometric isomorphism of the
duals of direct sums, quotient spaces, and subspaces of normed
spaces. The next section discusses reflexivity and includes Pettis's
theorem about the reflexivity of a closed subspace of a reflexive
space and many of its consequences. Section 1.12, devoted to separability,
includes the Banach-Mazur characterization of separable Banach
spaces as isomorphs of quotient spaces of \ell_1, and ends with
the characterization of separable normed spaces as the normed
spaces that are compactly generated so that the stage is set for
the introduction of weakly compactly generated normed spaces in
Section 2.8. This completes the basic material of Chapter 1.

The last section of Chapter 1, Section 1.13, is optional in
the sense that none of the material in the rest of the book outside
of other optional sections depends on it. This section contains
a number of useful characterizations of reflexivity, including
James's theorem. Some of the more basic of these are usually obtained
as corollaries of the Eberlein-Smulian theorem, but are included
here since they can be proved fairly easily without it. The most
important of these basic characterizations are repeated in Section
2.8 after the Eberlein-Smulian theorem is proved, so this section
can be skipped without fear of losing them. The heart of the section
is a proof of the general case of James's theorem: *A Banach
space is reflexive if each bounded linear functional x* on the
space has the property that the supremum of |x*| on the closed
unit ball of the space is attained somewhere on that ball.*
The proof given here is a detailed version of James's 1972 proof.
While the development leading up to the proof could be abbreviated
slightly by delaying this section until the Eberlein-Smulian theorem
is available, there are two reasons for my not doing so. The first
is that I wish to emphasize that the proof is really based only
on the elementary metric theory of Banach spaces, not on arguments
involving weak compactness, and the best way to do that is to
give the proof before the weak topology has even been defined
(though I do cheat a bit by defining weak sequential convergence
without direct reference to the weak topology). The second is
due to the reputation that James's theorem has acquired as being
formidably deep. The proof is admittedly a bit intricate, but
it is entirely elementary, not all that long, and contains some
very nice ideas. By placing the proof as early as possible in
this book, I hope to stress its elementary nature and dispel a
bit of the notion that it is inaccessible.

**Chapter 2** deals with the weak topology of a normed space
and the weak* topology of its dual. The first section includes
some topological preliminaries, but is devoted primarily to a
fairly extensive development of the theory of nets, including
characterizations of topological properties in terms of the accumulation
and convergence of certain nets. Even a student with a solid first
course in general topology may never have dealt with nets, so
several examples are given to illustrate both their similarities
to and differences from sequences. A motivation of the somewhat
nonintuitive definition of a subnet is given, along with examples.
The section includes a short discussion of topological groups,
primarily to be able to obtain a characterization of relative
compactness in topological groups in terms of the accumulation
of nets that does not always hold in arbitrary topological spaces.
Ultranets are not discussed in this section, since they are not
really needed in the rest of this book, but a brief discussion
of ultranets is given in Appendix D for use by the instructor
who wishes to show how ultranets can be used to simplify certain
compactness arguments.

Section 2.2 presents the basic properties of topological vector spaces and locally convex spaces needed for a study of the weak and weak* topologies. The section includes a brief introduction to the dual space of a topological vector space, and presents the versions of the Hahn-Banach separation theorem due to Mazur and Eidelheit as well as the consequences for locally convex spaces of Mazur's separation theorem that parallel the consequences for normed spaces of the normed space version of the Hahn-Banach extension theorem.

This is followed by a section on metrizable vector topologies.
This section is marked optional since the topologies of main interest
in this book are either induced by a norm or not compatible with
any metric whatever. An F-space is defined in this section to
be a topological vector space whose topology is compatible with
a complete metric, without the requirement that the metric be
invariant. Included is Victor Klee's result that every invariant
metric inducing a topologically complete topology on a group is
in fact a complete metric, which has the straightforward consequence
that every F-space, as defined in this section, actually has its
topology induced by a complete *invariant* metric, and thereby
answers a question of Banach. The versions of the open mapping
theorem, closed graph theorem, and uniform boundedness principle
valid for F-spaces are given in this section.

Section 2.4 develops the properties of topologies induced by
families of functions, with special emphasis on the topology induced
on a vector space *X* by a subspace of the vector space of
all linear functionals on *X* .

The study of the weak topology of a normed space begins in earnest in Section 2.5. This section is devoted primarily to summarizing and extending the fundamental properties of this topology already developed in more general settings earlier in this chapter, and exploring the connections between the weak and norm topologies. Included is Mazur's theorem that the closure and weak closure of a convex subset of a normed space are the same. Weak sequential completeness, Schur's property, and the Radon-Riesz property are studied briefly.

Section 2.6 introduces the weak* topology of the dual space
of a normed space. The main results of this section are the Banach-Alaoglu
theorem and Goldstine's theorem. This is followed by a section
on the bounded weak* topology of the dual space of a normed space,
with the major result of this section being the Krein-Smulian
theorem on weakly* closed convex sets: *A convex subset C of
the dual space X* of a Banach space X is weakly* closed if and
only if the intersection of C with every positive scalar multiple
of the closed unit ball of X* is weakly* closed.*

Weak compactness is studied in Section 2.8. It was necessary to delay this section until after Sections 2.6 and 2.7 so that several results about the weak* topology would be available. The Eberlein-Smulian theorem is obtained in this section, as is the result due to Krein and Smulian that the closed convex hull of a weakly compact subset of a Banach space is itself weakly compact. The corresponding theorem by Mazur on norm compactness is also obtained, since it is an easy consequence of the same lemma that contains the heart of the proof of the Krein-Smulian result. A brief look is taken at weakly compactly generated normed spaces.

The goal of optional Section 2.9 is to obtain James's characterization of weakly compact subsets of a Banach space in terms of the behavior of bounded linear functionals. The section is relatively short since most of the work needed to obtain this result was done in the lemmas used to prove James's reflexivity theorem in Section 1.13.

The topic of Section 2.10 is extreme points of nonempty closed convex subsets of Hausdorff topological vector spaces. The Krein-Milman theorem is obtained, as is Milman's partial converse of that result.

Chapter 2 ends with an optional section on support points and subreflexivity. Included are the Bishop-Phelps theorems on the density of support points in the boundaries of closed convex subsets of Banach spaces and on the subreflexivity of every Banach space.

**Chapter 3** contains a discussion of linear operators
between normed spaces far more extensive than the brief introduction
presented in Section 1.4. The first section of the chapter is
devoted to adjoints of bounded linear operators between normed
spaces. The second focuses on projections and complemented subspaces,
and includes Whitley's short proof of Phillips's theorem that
*c*_0 is not complemented in \ell_\infty.

Section 3.3 develops the elementary theory of Banach algebras and spectra, including the spectral radius formula, primarily to make this material available for the discussion of compact operators in the next section but also with an eye to the importance of this material in its own right.

Section 3.4 is about compact operators. Schauder's theorem relating the compactness of a bounded linear operator to that of its adjoint is presented, as is the characterization of operator compactness in terms of the bounded-weak*-to-norm continuity of the adjoint. Riesz's analysis of the spectrum of a compact operator is obtained, and the method used yields the result for real Banach spaces as well as complex ones. The Fredholm alternative is then obtained from this analysis. Much of the rest of the section is devoted to the approximation property, especially to Grothendieck's result that shows that the classical definition of the approximation property in terms of the approximability of compact operators by finite-rank operators is equivalent to the common modern definition in terms of the uniform approximability of the identity operator on compact sets by finite-rank operators. The section ends with a brief study of the relationship between Riesz's notion of operator compactness and Hilbert's property of complete continuity, and their equivalence for a linear operator whose domain is reflexive.

The final section of Chapter 3 is devoted to weakly compact operators. Gantmacher's theorem is obtained, as well as the equivalence of the weak compactness of a bounded linear operator to the weak*-to-weak continuity of its adjoint. The Dunford-Pettis property is examined briefly in this section.

The purpose of **Chapter 4** is to investigate Schauder
bases for Banach spaces. The first section develops the elementary
properties of Schauder bases and presents several classical examples,
including Schauder's basis for *C*[0,1] and the Haar basis
for *L*_*p*[0,1] when 1\le *p*<\infty . Monotone
bases and the existence of basic sequences are covered, and the
relationship between Schauder bases and the approximation property
is discussed.

Unconditional bases are investigated in Section 4.2. Results
are presented about equivalently renorming Banach spaces with
unconditional bases to be Banach algebras and Banach lattices.
It is shown that neither the classical Schauder basis for *C*[0,1]
nor the Haar basis for *L*_1[0,1] is unconditional.

Section 4.3 is devoted to the notion of equivalent bases and
applications to finding isomorphic copies of Banach spaces inside
other Banach spaces. Characterizations of the standard unit vector
bases for *c*_0 and \ell_1 are given. Weakly unconditionally
Cauchy series are examined, and the Orlicz-Pettis theorem and
Bessaga-Pelczynski selection principle are obtained.

The properties of the sequence of coordinate functionals for
a Schauder basis are taken up in Section 4.4, and shrinking and
boundedly complete bases are studied. The final section of Chapter
4 is optional and is devoted to an investigation of James's space
*J*, which was the first example of a nonreflexive Banach
space isometrically isomorphic to its second dual.

**Chapter 5** focuses on various forms of rotundity, also
called strict convexity, and smoothness. The first section of
the chapter is devoted to characterizations of rotundity, its
fundamental properties, and examples, including one due to Klee
that shows that rotundity is not always inherited by quotient
spaces. The next section treats uniform rotundity, and includes
the Milman-Pettis theorem as well as Clarkson's theorem that the
*L*_*p* spaces such that 1<*p*<\infty are
uniformly rotund. Section 5.3 is devoted to generalizations of
uniform rotundity, and discusses local uniform rotundity, weak
uniform rotundity, weak* uniform rotundity, weak local uniform
rotundity, strong rotundity, and midpoint local uniform rotundity,
as well as the relationships between these properties.

The second half of Chapter 5 deals with smoothness. Simple smoothness is taken up in Section 5.4, in which the property is defined in terms of the uniqueness of support hyperplanes for the closed unit ball at points of the unit sphere and then characterized by the Gateaux differentiability of the norm and in several other ways. The partial duality between rotundity and smoothness is examined, and other important properties of smoothness are developed. Uniform smoothness is the subject of the next section, in which the property is defined using the modulus of smoothness and characterized in terms of the uniform Frechet differentiability of the norm. The complete duality between uniform smoothness and uniform rotundity is proved. Frechet smoothness and uniform Gateaux smoothness are examined in the final section of the chapter, and Smulian's results on the duality between these properties and various generalizations of uniform rotundity are obtained.

**Appendix A** includes an extended description of the prerequisites
for reading this book, along with a very detailed list of the
changes that must be made to the presentation in Chapter 1 if
this book is to be used for an undergraduate topics course in
Banach space theory. **Appendices B** and **C** are included
to support such a topics course. They are, respectively, a list
of the properties of metric spaces that should be familiar to
a student in such a course and a development of \ell_*p*
spaces from basic principles of analysis that does not depend
on the theory of *L*_*p* spaces. **Appendix D** is
a discussion of ultranets that supplements the material on nets
in Section 2.1.