Functional Analysis and Numerical Mathematics

482 Pages·1966·17.987 MB·English

by  Lothar Collatz (Auth.)

FUNCTIONAL
ANALYSIS
AND NUMERICAL
MA THEMA TICS
by LOTHAR COLLATZ
DEPARTMENT OF MATHEMATICS
HAMBURG UNIVERSITY
HAMBURG, GERMANY
Translated by HANS JÖRG OSER
NATIONAL BUREAU OF STANDARDS
WASHINGTON, D.C.
AND
CATHOLIC UNIVERSITY OF AMERICA
WASHINGTON, D.C.
1966
ACADEMIC PRESS New York and London
@
First published in the German Language under the title
Funktionanalysis und Numerische Mathematik
and copyrighted in 1964 by Springer-Verlag, Berlin-Göttingen-Heidelberg
COPYRIGHT © 1966, BY ACADEMIC PRESS INC.
ALL RIGHTS RESERVED.
NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM,
BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT
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PRINTED IN THE UNITED STATES OF AMERICA
Translator's Note
This book is a translation of the German edition published in
1964. No new material is included except for a supplement of English
titles in the bibliography which is intended to facilitate the entry
into the literature in the field for the English speaking reader.
It is with great pleasure that the translator acknowledges help and
advice from many of his colleagues. Dr. J. Wlocka, from the University
of California, reviewed the entire manuscript and had many sugges­
tions for improvements.
The translator is grateful to the Mathematics Research Group at
Wright Patterson Air Force Base, in particular Drs. G. Blanche,
D. G. Shankland, C. Keller, B. Mond, P. Nicolai, and others who
painstakingly went over the manuscript and pointed out many
omissions, misprints, and places where improvements were in order.
The author himself took active interest in the progress of the
translation and readily agreed to a number of suggested changes or
additions.
Finally a tribute to the publisher who showed infinite patience in
accepting last minute changes in the text. My thanks to all who
made it possible for this translation to appear.
Washington, D.C. HANSJÖRG OSER
June, 1966
Preface to the German Edition
This book does not claim to be a textbook of either functional
analysis or numerical analysis; its purpose is merely to point out the
structural changes which numerical analysis has undergone, on the
one hand as a result of the widespread use of large electronic computers,
on the other hand through the development of abstract methods.
The resulting picture of numerical analysis is quite different from the
one of ten or twenty years ago. Just as in other areas of mathematics,
a strong trend towards abstraction is apparent in numerical analysis.
But at the same time, the boundaries between different mathematical
disciplines disappear. It is for this reason that it is so difficult at
present to decide whether functional analysis belongs to pure or to
so-called applied mathematics. Functional analysis is a foundation
for large segments of the two disciplines mentioned above and the
author would be delighted to find that this book contributes to
showing how absurd the distinction between "pure" and "applied"
mathematics actually is; there is really no boundary that separates
the two, there is only one mathematics, of which analysis, topology,
algebra, numerical analysis, probability theory, etc., are merely some
overlapping areas.
This book does not pretend to be complete. In recent years so many
applications of functional analysis to numerical analysis have been
pointed out that it would be far beyond the scope of this book to
mention them all. This is merely an attempt to select a few applica­
tions and thus stimulate those who enjoy theory as well as those who
prefer numerical work. The proponents of theory are sometimes
unaware of the fascinating applications that their theory permits and
of the important problems in numerical analysis which have yet to
Vll
Vlll PREFACE TO THE GERMAN EDITION
be solved (and there are a great many problems that still remain
untackled). We are at the beginning of a great development. Technical
and physical problems are often so complicated that yesterday's and
today's mathematics cannot produce satisfactory answers, in particular
for the increasing number of nonlinear problems which begin to
dominate the scene. On the other hand, those who solve problems
on electronic computers are sometimes not aware that enough
mathematical tools exist to estimate, for example, the accuracy of the
solution and to ascertain the number of valid figures in the computer
results.
David Hubert started an extremely fruitful development by his
idea of function spaces. "Hubert spaces" proved fundamental for
wide areas of applications; in particular, the wide field of eigenvalue
problems should be mentioned which is so important for theoretical
physics. It turned out, however, that the nonlinear problems which
became more and more important required more general spaces and
it was Stephan Banach who pointed the way here. Subsequently,
still more general spaces became subject to investigation. Generality
shall here be carried only as far as the applications in numerical
analysis make it desirable and necessary; indications show that the
pseudometric spaces introduced by Kurepa at present seem to be
the most important generalization of the hitherto considered spaces
for the purposes of numeral analysis. An attempt has been made to
use the weakest possible assumptions (on norms and distances, for
example).
It is one of the aims of the present book to speak to the proponents
of the applications by purposely allowing some redundancy, in
particular, in the first few chapters, certainly too redundant for the
theorist; but the author finds it important that the presentation be
such that the interested physicist and engineer can read it, thus
creating a desire to apply theory, even at the danger of having the
theorist reject the presentation as being too trivial. Following the
technical literature, both customary symbols for functions are being
used, with and without argument, for example, / as well as f(x).
Chapter I contains largely theory, while Chapters II and III deal
with numerical applications, although numerics are mentioned in
Chapter I and Chapters II and III cannot eschew all theory. This
shows, perhaps, how closely interwoven numerics and theory already
are.
PREFACE TO THE GERMAN EDITION ix
The author pondered a long while on whether he should include a
proof of Schauder's fixed point theorem or point out other references;
in view of a forthcoming publication of J. Schröder the author chose
the second alternative, certainly to the regret of many readers. Except
for this theorem, the author endeavored to prove all necessary
theorems and refrained from citing other references. Some textbooks
on functional analysis contain a detailed theory of the Lebesgue
integral; this has been omitted, since there are numerous excellent
texts on Lebesgue integrals, but also for the reason that, from the
general point of view of the applications, the Banach space C[B] of
continuous functions (in Chapter I, Section 4.3) is certainly of far
greater importance than the Hubert space L2(B) (of Chapter I,
Section 4.3).
This book developed from notes on lectures which the author gave
during the last ten years at the University of Hamburg. He owes a
great deal to conversations and discussions with the members of the
Institute and to the Seminars conducted together with L. Schmetterer
and H. Bauer. He also thanks the many members of the Institute
for carrying out the computations for the examples, in particular
those on the electronic computer; Mr. Hadeler deserves credit for
his help in procuring the biographic material; special thanks go to
Dr. Erich Bol, Dr. Siegfried Gruber, Dr. Werner Krabs, Hermann
MierendorfF, and Roland Wäis for the troublesome and very careful
proof reading and for many suggestions for improvements; finally
tributes are due to Springer-Verlag for always agreeing with my wishes
in the most understanding way.
Hamburg, Germany LOTHAR COLLATZ
Spring, 1964
Notation
From Set Theory
If M is a set of elements, then the symbol feM means that / is
an element of M. f $ M means that / does not belong to M.
Φ designates the empty set which does not contain any elements.
It is also called the null set.
A set M with elements x having property A shall be represented
by {x | A}; for example, {x | 1 < x ^ 2} is the set of all real numbers
on the closed interval [1,2] and {x> y \ x2 + y2 < 1, x > 0} is the
set of all points in the x-y plane which belong to the closed semi­
circle x2 + y2 < 1, x > 0.
Μ Π M is the intersection of the two sets M and M , which is
λ 2 x 2
the set of all elements that belong to both M and M .
1 2
M U M is the union of the two sets M and M , which is the
x 2 x 2
set of all elements belonging to either M or M .
x 2
If 5 is a (proper or improper) subset of M, then M — 5 is the
complementary set of S with respect to M, that is, the set of all
elements of M that do not belong to S.
Let M and N be two sets, then MDN or NC M means that all
elements of iV are also elements of M. M D N or N C M expresses
the fact that M D N> but that there exists at least one element in M
which does not belong to iV.
The Cartesian (or direct) product of two sets is defined as
M x N = {(m, n)\me M, n e JV},
that is, the set of all pairs (m, n) where m belongs to M and n belongs
to N.
XIX
XX NOTATION
From Various Fields in Mathematics
R is the w-dimensional space of points with coordinates
n
x, x , ..., x . We distinguish the real space R from the complex
x 2 n n
space R depending on whether the x. are real or complex numbers.
n {
A subscript of a function symbol indicates the partial derivative
of that function with respect to the variable in the subscript; for
example, for the function u(x, y) we write u for dujdx or u for
x xy
82u/dx dy. V2 = Σ™ d2\dx\ is the Laplacian operator of the inde­
=1
pendent variables x , x , ..., x .
1 2 n
Let A be a matrix with elements a (j = 1,2, ..., n\k — 1, 2, ..., n).
jk
Ä is called the conjugate matrix of A with elements ä , and A' is
jk
the transpose of A with elements a . The transpose of a single-
kj
column matrix (or column vector) is a row vector x'. A* = {ä ) = Ä'
kj
is the adjoint matrix of A = (a ).
jk
The Landau Symbol
f(x) — 0(£(#)) expresses the fact that for the two functions f(x)
and g(x), which are defined on a domain Z), and where g(x) > 0,
there exists a constant K with the property \f(x)\ < ^(^) for all
Abbreviations Used in This Book
Names followed by a number in brackets, such as [58] or [58a],
refer to the references at the end of this book, the number indicating
the year of publication [(19)58].
CHAPTER I
Foundations of
Functional Analysis and Applications
L TYPICAL PROBLEMS IN NUMERICAL MATHEMATICS
LI Some General Concepts
A systematic introduction of the mathematical concepts will be
found in Sections 2 through 6. This introductory section will merely
serve to mention a few concepts that will be needed to describe the
problems with which this book deals.
A "space" R is a set of elements/, £,... . In the applications these
elements may be real numbers, complex numbers, vectors, matrices,
functions of one or more variables, systems of such functions, or
systems consisting of any number of the above-mentioned examples,
such as pairs of a function and a real number, and so on.
Ordinarily the spaces that occur in the applications are linear and
hence, we shall consider linear spaces only.
Definition. A linear space is characterized by the following
properties:
1. An operation is defined, which we call addition, which follows
the rules of ordinary addition (explained more precisely in Section 2.4);
that is, when / and g are elements of R, f -f g belongs also to R, and
there exists a "null element" Θ in R with the property Θ +/ = /,
for all/ei?.
1
2 I. FOUNDATIONS OF FUNCTIONAL ANALYSIS AND APPLICATIONS
2. A multiplication is defined between the elements / and scalars c
of a field F. This multiplication obeys the rules of ordinary vector
algebra, that is, with f e R and c eF the product cf belongs also
y
to R. The field F is usually the field of rational, real, or complex
numbers.
Now we consider transformations (operators, mappings) T which
associate certain elements / of the "original space" R uniquely with
elements A of a linear space /?*, called the "image space." It will
frequently be the case that R = /?*. In the case when i?* is a number
space, T associates each/with a number and jTis called a "functional."
If /?* contains real numbers only, then T is called a real functional.
An operator T is called linear when T is defined for all / e R
(or at least on a linear manifold, see Section 2.4.), and when
Γ(Ί/ι + cjt) = Tf + c Tf (1.1)
Cl x 2 2
holds for all elements f,feR and all c, c e F.
x 2 x 2
In all other cases T is called a nonlinear operator.
Five categories of typical problems will be presented in
Sections 1.2-1.6. This classification is not exhaustive. Furthermore,
some problems may belong to several categories. Problems from
probability theory, statistics, and related areas which also belong to
numerical mathematics are not mentioned at all. These disciplines
have developed so extensively on their own that they are not treated
in this book. We refer the reader to textbooks in the bibliography.
1*2 Solutions of Equations
Let the unknown quantity u be an element of a given linear space R.
T (and S) are given linear or nonlinear operators. Three types of
equations can be distinguished:
Tu = u (1.2)
(the image elements are in the same space R, fixed elements are
sought);
Su = S (1.3)