Table Of ContentMATHEMATICAL
METHODS
JACOB KOREVAAR
UNIVERSITY OF CALIFORNIA, SAN DIEGO
LA JOLLA, CALIFORNIA
Volume 1
Linear algebra / Normed spaces / Distributions / Integration
ACADEMIC PRESS New York and London
COPYRIGHT © 1968, BY ACADEMIC PRESS INC.
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PREFACE
The volumes on mathematical methods are intended for students in the
physical sciences, for mathematics students with an interest in applications,
and for mathematically oriented engineering students. It has been the author's
aim to provide
(1) Many of the advanced mathematical tools used in applications;
(2) A certain theoretical mathematical background that will make most
other parts of modern mathematical analysis accessible to the student of
physical science, and that will make it easier for him to keep up with future
mathematical developments in his field.
From a mathematical point of view the presentation in these two volumes
is fairly rigorous, but certainly not abstract. If the student of physical science
finds parts of the material somewhat "theoretical," he should realize that
the power of modern mathematics derives in large measure from its abstract-
ness: It is because of its generality that mathematics is so widely applicable.
If the student of mathematics finds parts of the material somewhat concrete,
he should realize that mathematics is useful largely because it enables one to
make calculations.
For many years, the author has taught an intensive beginning graduate
course for students in the physical sciences and applied mathematics. The
present volume contains the material covered in the first semester or quarter
of that course. This introductory and relatively general material serves to
prepare the student for such subjects as orthogonal series, linear operators
in Hilbert space, integral equations, Sturm-Liouville problems, and partial
differential equations. All but the last topic will be the subject of the second
volume.
Naturally, Volume 1 can also be used in a leisurely two semester or two
quarter course for advanced undergraduate students. The principal prerequi
site would be a year of advanced calculus; in addition, some knowledge of
elementary linear algebra and elementary differential equations would be
desirable.
v
vi PREFACE
Chapter One deals with relevant topics in linear algebra. The emphasis is
on an understanding of the basic concepts of vector space and linear trans
formation. The treatment compares to that in the more sophisticated under
graduate texts on linear algebra: It is largely coordinate-free so that it applies
to infinite-dimensional as well as finite-dimensional situations. Matrices are
introduced as concrete representations of linear transformations, deter
minants are defined in terms of linear operators which gives them intuitive
geometric meaning. Subjects such as abstract systems of linear equations,
eigenvalue problems, matrix representation of linear operators, and the alge
braic theory of tensors are carried somewhat further than in the usual texts
on linear algebra.
Chapter Two provides an introduction to functional analysis. It begins
with a detailed discussion of the many different kinds of convergence for
sequences of functions which occur in practice, so that a motivation for
various metrics and norms is available. A basic theme is provided by the
construction called completion which is applied to a number of function
spaces. It is shown how the Lebesgue integrable functions may be ob
tained very simply by completion of a space of step functions. The normed
vector space L2, a concrete example of a Hilbert space, is also constructed
by completion. Another application of the method of completion leads to
the distributions or generalized functions which are becoming a more and
more important tool in the physical sciences and applied mathematics.
Distributions are discussed from the beginning both as (generalized)
limits of fundamental sequences of functions and as continuous linear func-
tionals. This method which has not been published before combines the ad
vantages of the original approach of S. L. Sobolev and L. Schwartz with
those of the more intuitive elementary theories developed by the author,
J. Mikusinski and R. Sikorski, and G. Temple. Delta sequences, or funda
mental sequences belonging to the delta distribution, provide a unifying
theme for a large class of theorems on convergence and approxima
tion.
Chapter Three deals with integration theory. The properties of Lebesgue
integrable functions are developed further. The emphasis is on how to operate
with Lebesgue integrals, and thus on theorems dealing with such topics as
termwise integration, inversion of the order of integration, differentiation
under the integral sign, and change of variables. Stieltjes integrals are intro
duced for the discussion of line integrals. Among the applications is a fairly
general form of Green's theorem in the plane not found in other books
which leads directly to the general form of Cauchy's theorem for line integrals
in the complex domain. There are also applications to potential theory. The
brief and to some extent original sketch of complex analysis at the end of the
chapter will be sufficient for the applications in Volume 2.
PREFACE Vil
In each chapter, some of the more difficult or less central topics and proofs
have been starred. Of certain proofs only an outline has been included while
a few have been omitted altogether. In these cases there are always one or
two references to the literature. Each chapter has its own bibliography;
references within the text are by name of author.
The present form of the book owes a great deal to the questions and re
marks of the many students who have taken the course. I wish to thank in
particular my former Ph.D. students Judith M. Elkins, Maynard D. Thomp
son, and Gilbert G. Walter who commented on early versions of some of the
material; remarks by certain anonymous referees have also been very helpful.
It is a pleasure to acknowledge the assistance by members of the mathe
matics office staffs, both at the University of Wisconsin and the University
of California, San Diego, who helped prepare successive approximations to
the final manuscript in the form of lecture notes. Thanks are due also to the
staff of Academic Press. Last but not least, I wish to mention the special
support of my wife Jopie, who suffered it all (if not altogether in silence).
La Jolla, California Jacob Korevaar
December, 1967
ONE
ALGEBRAIC THEORY
OF VECTOR SPACES
The basic concepts introduced in this chapter are the notions of vector
space and linear transformation.
A vector space or linear space V may be described briefly as a collection
of elements with the following properties :
(i) If x and y are any two elements of V then V also contains an element
which may be called the sum x + y of x and y.
(ii) If x is any element of V and λ an arbitrary scalar (the scalars are
usually the real or the complex numbers) then V also contains an
element which may be called the scalar multiple λχ.
The ordinary vectors in ordinary space, or in " ordinary " «-dimensional
space, form a vector space. Important vector spaces of functions are given
by the continuous functions on an interval, the integrable functions, and the
n times continuously differentiable functions.
A linear transformation L is a transformation from one vector space into
another (or of a vector space into itself) which commutes with addition and
multiplication by scalars. Thus, denoting by Lv the vector which L assigns to
the vector v,
L(x + y) = Lx + Ly, LÀx = XLx.
1
2 1. ALGEBRAIC THEORY OF VECTOR SPACES
A simple example is given by ordinary differentiation; more generally,
every linear ordinary or partial differential operator defines a linear trans
formation. Integration over a fixed interval defines a linear transformation;
integral transformations such as the Laplace and the Fourier transformation
are linear. Matrix products AxT (where A is a matrix, xT a column vector)
can be used to define linear transformations from one finite dimensional
vector space into another.
Many so-called linear problems in mathematics can be written in the
abstract form
Lx = z,
where L is a known linear transformation, z a given vector, and the vector x
is to be determined. Examples are given by systems of linear algebraic
equations, linear ordinary and partial differential equations and systems of
such equations, and linear boundary value problems for ordinary and partial
differential equations.
The present chapter develops the algebraic machinery required to deal
with linear problems. Coordinate-free definitions will be given for all im
portant concepts. This is essential in the case of the usual function spaces,
since in these infinite dimensional spaces there are no natural or simple
coordinate systems. In the finite dimensional case the abstract approach is
valuable because it provides additional insight, especially if the reader is
already familiar with the classical theory of vector spaces of «-tuples, matrices,
and determinants. In this book, matrices are defined as representations of
linear transformations; the determinant of a matrix is defined in terms of the
determinant of a linear transformation (which has a simple geometric
meaning).
It will turn out that the algebraic tools are adequate as long as one deals
with finite dimensional problems, even in infinite dimensional spaces (cf.
Secs. 9.7 and 10.4). However, for infinite dimensional problems in function
spaces the purely algebraic methods no longer suffice. The study of such
problems requires additional notions such as convergence, metric, norm, and
scalar product, which will be introduced in later chapters. Linear transfor
mations in normed vector spaces and scalar product spaces will be studied
in Vol. 2. The special properties of symmetric matrices and quadratic forms
will be discussed only at that time.
A great many textbooks for undergraduate algebra courses contain a
good introduction to the algebraic theory of vector spaces; we list several in
the Bibliography at the end of the chapter. One book, devoted entirely to
vector spaces, stands out: it is P. R. Halmos' Finite-dimensional Vector
Spaces.
1. VECTOR SPACES 3
1. VECTOR SPACES
In a discussion of vector spaces or linear spaces it is convenient to think
of the following model. Consider the collection of all directed line segments
OP in some finite or perhaps infinite dimensional space which start at a fixed
point O. There will be only two operations defined on this collection of
directed line segments, namely, addition and multiplication by scalars. The
sum of any two directed line segments OP and OQ is given by the directed
diagonal OR of the parallelogram on OP and OQ (Fig. 1-1). The product of
the (real) scalar λ and the directed line segment OP is given by the directed
line segment OS defined as follows. The length of OS is equal to the product
of the length of OP by \λ\; the direction of OS is the same as that of OP if
λ > 0 and is opposite that of OP if λ < 0. (In Fig. 1-2, OS = f OP, and
OT= — i<9P.)The product of the scalar 0 and the directed segment OP is
the zero segment 00 of undetermined direction.
O
Fig. 1-1 Fig. 1-2
Actually the mathematical concept of a vector space involves fewer
notions than the above model suggests. It is essential that one have a collection
of objects on which an addition and a multiplication by scalars are defined,
but lengths and directions play no role in the abstract definition. [One intro
duces lengths of vectors only in the special case of a normed vector space
(Chapter 2), and angles only in the even more special case of a scalar product
space (Vol. 2).]
In "ordinary" «-dimensional space one could define sums and scalar
multiples in terms of components. However, this would presuppose a co
ordinate system and in many "large" spaces there exist no simple coor
dinate systems.
1.1. Definition
Before we give a formal definition of the notion of " vecto rspace " we
have to say something about the concept of the associated " scalars." Any
4 1. ALGEBRAIC THEORY OF VECTOR SPACES
(commutative) field #" can play the role of a set of scalars for a vector space.
However, in this book the set of scalars will almost always be either the
field ^ of the real numbers, or the field ^ of the complex numbers.
A (commutative) field is a mathematical system in which one can form
the sum, the difference, the product, and the quotient of any two elements
without going outside the given system (except that one cannot divide by 0).
Besides a zero element every field contains an element 1^0. (There exists a
field consisting of just two elements 0 and 1 ; its basic rule is 1 + 1=0.)
The operations of addition and multiplication must satisfy familiar rules
such as commutativity and associativity.
Now let 3F be a given field. We will usually denote the elements of #",
the scalars, by Greek letters.
A vector space V "over ^" is a collection of elements—referred to
as vectors—with a rule for addition and a rule for multiplication by scalars.
These rules must satisfy the following axioms.
ADDITION AXIOMS
(i) To every pair of vectors x and y in V, where y need not be different
from x, there is a unique vector z in V which is called the sum
x + y of x and y.
(ii) x + y = y + x for every x and y in V (commutative law).
(iii) x + (y + z) = (x + y) + z for every x, y, and z in V (associative
law).
(iv) There is a unique zero vector in V, that is, a vector 0 such that
x + 0 = x for every x in V.
(v) To every x in V there is a unique vector ( — x) in V such that
JC + (-X) = 0.
AXIOMS FOR MULTIPLICATION BY SCALARS
(vi) To every element (scalar) λ in 2F and every vector x in V there
is a unique vector w in V which is called the scalar multiple λχ of x.
(vii) λ(μχ) = (λμ)χ for every λ and μ in 3F and every x in V.
(viii) 1 · x = x for every x in V.
(ix) 0 · x = 0 for every x in K
DISTRIBUTIVE LAWS
(x) λ(χ + y) = λχ + Ày for every λ in «^* and every x and }> in V.
(xi) (Λ. + μ)χ = λχ + μχ for every A and μ in J^ and every x in K.
1. VECTOR SPACES 5
The axioms are not independent. For example, (ix) can be derived from
the other axioms, but it is convenient to have it listed as a rule. One can
likewise prove that (— l)x is the vector ( — x) whose existence was postulated
in (v). Note that if x and y are any two vectors in V, then V must contain every
linear combination λχ + μγ with λ and μ in &'. We will usually write x — y
for * + (->>).
If (and only if) !F is the field of the real numbers we call V a real vector
space \ if 3F is the field of the complex numbers, we call V a complex vector
space.
1.2. Vector Spaces of «-Tuples
Consider the vectors x — OP in the ordinary plane which start at a given
point O. After we introduce a coordinate system in the plane with origin O,
every vector x can be described by an ordered pair (ξ ξ ) of real numbers,
ΐ9 2
the components of x. The sum of the vectors x = (ξ ξ ) and y = (η η ) is
ί9 2 ΐ9 2
the vector z with components ξ + η , ξ + η (Fig. 1-3). The scalar multiple
1 1 2 2
λχ is the vector w with components λξ λξ . The zero vector is given by the
ΐ9 2
pair (0, 0), the vector ( — x) by the pair ( — ξ —ξ ).
ί9 2
One can similarly describe the vectors x = OP in ordinary space by
ordered triples (ξ ζ , ξ ) of real numbers.
ί9 2 3
After this geometrical introduction we give a purely algebraic definition
of the vector space 8% , where n may be any positive integer. The elements
n
