Table Of ContentSTUDIES IN FUNCTIONAL
ANALYSIS
MAA STUDIES IN MATHEMATICS
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Studies in Mathematics
The Mathematical Association of America
Volume 1: STUDIES INMODERN ANALYSIS
editedby R. C. Buck
Volume 2: STUDIES IN MODERNALGEBRA
editedbyA. A. Albert
Volume 3: STUDIESINREALANDCOMPLEXANALYSIS
editedby I. I. Hirschman,Jr.
Volume : STUDIES IN GLOBALGEOMETRYANDANALYSIS
editedby S. S. Chern
Volume : STUDIESINMODERNTOPOLOGY
editedby P. J. Hilton
Volume : STUDIES INNUMBERTHEORY
editedby W. J. LeVeque
Volume : STUDIES IN APPLIEDMATHEMATICS
editedbyA. H. Taub
Volume STUDIES IN MODELTHEORY
editedbyM. D. Morley
Volume 9: STUDIES INALGEBRAICLOGIC
editedbyAubertDaigneault
Volume 10: STUDIESIN OPTIMIZATION
editedby G. B. DantzigandB. C. Eaves
Volume 11: STUDIESIN GRAPHTHEORY, PARTI
editedby D. R. Fulkerson
Volume 12: STUDIESINGRAPHTHEORY, PARTII
editedby D. R. Fulkerson
Volume 13: STUDIESINHARMONICANALYSIS
editedbyJ. M. Ash
Volume 14: STUDIESIN ORDINARYDIFFERENTIALEQUATIONS
editedbyJackHale
Volume15: STUDIESIN MATHEMATICALBIOLOGY, PARTI
editedby S. A. Levin
Volume16: STUDIES IN MATHEMATICALBIOLOGY, PARTII
editedby S. A. Levin
Volume17: STUDIES IN COMBINATORICS
editedby G.-C. Rota
Volume 18: STUDIESIN PROBABILITYTHEORY
editedby Murray Rosenblatt
Volume 19: STUDIESIN STATISTICS
editedbyR. V. Hogg
Volume20: STUDIES INALGEBRAICGEOMETRY
editedbyA. Seidenberg
Volume21: STUDIESINFUNCTIONALANALYSIS
editedbyR. G. Bartle
Robert G. Bartle
University ofIllinois, Urbana-Champaign
F. F. Bonsall
University ofEdinburgh, Scotland
E. W. Cheney
University ofTexas, Austin
J. Duncan
University ofStirling, Scotland
WilliamB. Johnson
The Ohio State University, Columbus
R. R. Phelps
University ofWashington
Hi. H. Schaefer
University ofTiibingen
FederalRepublicofGermany
Studies in Mathematics
Volume21
STUDIES IN FUNCTIONAL ANALYSIS
R. G. Bartle, editor
University ofIllinois at Urbana-Champaign
Publishedanddistributedby
The Mathematical Association of America
© 1980 by
The MathematicalAssociationofAmerica (Incorporated)
Library ofCongress Catalog CardNumber80-81042
CompleteSetISBN0-88385-100-8
Vol.21 ISBN0-88385-121-0
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Currentprinting(lastdigit):
10987654321
INTRODUCTION
In the introduction to his article “Preliminaries to functional
analysis” that appeared in Volume | of these Studies in Mathemat-
ics (1962), CasparGoffman wrote: “Functional analysis, briefly, is
that branch of mathematics in which elements of a given class of
functions are considered to be points of an appropriate infinite
dimensional space. In this way, various theories of functions, real
and complex, are seen from either a geometric or a point-set
theoretic view, often achieving new levels of unification.”
While this definition of the term “functional analysis” is by no
means perfect, it is probably as good as anyone is likely to
formulate. To be sure, the objects of study in functional analysis
are not always functions (in a natural way, at least) or operators
on them; but they generally are, and it is this fact that puts
functional analysis firmly in the camp of analysis, rather than that
of topology. It is true that the mathematicians who, about 1908,
laid the foundations of functional analysis, and among whom we
count Fréchet, F. Riesz, Hausdorff, and E. Schmidt, were also
active in the founding of point-set topology. However, from the
very beginning, what we nowthink of as functional analysis was
nourished by the rich analytic work that had been done in a
somewhat more classical manner by such giants as Volterra,
Fredholm, Hadamard, and Hilbert. F. Riesz, while not the first to
formulate axiomsclosely related to thoseof anormedlinearspace,
Vii
viii Robert G. Bartle
gave a beautifully ‘‘modern” discussion of a general form of the
Fredholm theory of compact operators in his 1918 Acta Mathe-
matica paper. With this paper, “functional analysis” was certainly
a reality.
By the end of the 1920's, the work of Lebesgue, Helly, Hahn,
Hildebrandt, Banach, and Schauder had developed the main re-
sults in the abstract theory of what we now call Banach spaces.
Meanwhile, von Neumann and Stone were working on the related
Hilbert space theory with an eye towardits application, especially
to differential equations. The year 1932 was surely a milestone in
the history of functional analysis, for in that year the books of
Banach and Stone were both published, and they remained the
definitive treatises, at least until the appearance of Hille’s “Func-
tional analysis and semi-groups” in 1948. But there had been an
explosive development of the field during that time. To mention
only some of the highlights, we cite: (i) the development of the
general theory of topological vector spaces initiated by
Kolmogorov (1934), von Neumann (1935), Dieudonné (1943),
Mackey (1946), Arens (1947), et al.; (ii) the fixed poimt theorems
of Schauder (1930), Tihonov (1935), Markov (1936), and Kakutani
(1938) and their application to differential equations by Leray and
Schauder (1934); (iii) the development of abstract integrals by
Bochner (1933), G. Birkhoff (1935), Dunford (1935, 1936),
Gel’fand (1938), Pettis (1938), et al.; (iv) the work of Stone (1937)
on the representation of Boolean rings, of M. and S. Krein (1940)
and Kakutani (1941) on Z- and M-spaces, and Gel’fand on
“normed rings” (now called “Banach algebras”); (v) the Krein-
Mil’man (1940) theorem on extreme points and the theorems of
Krein and Smulian (1939, 1940) on weak compactness and convex
sets; (vi) the operational calculus of E. R. Lorch (1942) and
Dunford (1943); and (vii) the work of Murray and von Neumann
(1936-1943) on rings of operators. Even this list, whichis intended
to be illustrative rather than exhaustive, does not suggest the
wealth of the applications that were beginning to be made of
functional analysis in other areas, such as Fourier analysis,
harmonic analysis, differential and integral equations, approxima-
tion theory, probability theory, ergodic theory, quantum
mechanics, and many other areas.
INTRODUCTION 1X
In this small volume we present to the reader five expository
accounts of some of the developments that have taken place
recently, Recognizing that it is not possible to give a full account
(and perhaps not even a balanced one) of the present state of the
field in such limited space, we have focused on several rather
diverse topics. We believe that the reader will find it fascina-
ting—as we do—to see how varied are the algebraic, analytic,
topological, and combinatorial techniques that are employed by
the authors of these articles. While these articles are directed
somewhat more toward the space than the operators on them, the
readerwill see a considerable interplay between these notions.
In the first article, Bonsall and Duncan give an account of
recent work using the notion of numerical range of operators in
Banachspaces. This tool, which at first glance seems too elemen-
tary to be promising, is seen—by using some shrewd geometrical
and analytic arguments—to have some surprising consequences,
among them, a proof of an abstract characterization of C*-alge-
bras.
In the second article, Cheney gives an overview of certain
questions in the theory of approximation. He showshow the desire
to settle certain “practical” questions concerning optimal ap-
proximation leads to a need to know more aboutthe geometrical
nature of certain Banach spaces.
Compactoperators have been with functional analysis since the
start. Some of the initial studies of compact operators in Hilbert
space were based on the fact that such operators can be approxi-
mated by finite rank operators, so it has long been a source of
embarrassment that the corresponding fact in Banach spaces was
not known until Enflo gave a counterexample, in 1973. Johnson
presents a description of how such a counterexample can be
constructed, using a “problems-with-hints” format. In doing so, he
leads the reader into a portion of Banach space structure theory, a
field that has been very active recently. While the reader will find
some of the problems easy, he will discover that others require
considerable combinatorial skill.
The Krein-Mil’man Theorem is one of the major results of
functional analysis. Recently, Choquet (1956) proved a result that
can be regarded as a refinement of the Krein-Mil’man theorem.In
x Robert G. Bartle
the fourth article, Phelps gives an exposition of Choquet’s theorem
and its applications. This article is an interesting mixture of
geometry andanalysis.
While the notion of order was studied in a functional analytic
setting by Kantorovié (1937) and others, the theory of ordered
spaces has remained somewhat outside of the mainstream of
functional analysis until recently. In the final article, Schaefer
gives a thorough accountof the theory of Banach lattices, includ-
ing a proof of the Kakutani-Krein representation theorems, and
an introduction to tensor products of Banach lattices. He gives a
numberof examples and applications of these important results.
While there are a number of important areas of functional
analysis that are not touched onatall in this study, we feel there is
enough variety to interest many readers andthatit gives a taste of
whathas been donerecently in functional analysis.
ROBERT G. BARTLE
