Functional_Analysis_Lecture_Notes-T_B_Ward.pdf

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Functional analysis lecture notes
T.B. Ward
Author address:
School of Mathematics, University of East Anglia, Norwich NR4
7TJ, U.K.
E-mail address: t.ward@uea.ac.uk
Course objectives
In order to reach the more interesting and useful ideas, we shall adopt a fairly
brutal approach to some early material. Lengthy proofs will sometimes be left
out, though full versions will be made available. By the end of the course, you
should have a good understanding of normed vector spaces, Hilbert and Banach
spaces, xed point theorems and examples of function spaces. These ideas will be
illustrated with applications to dierential equations.
Books
You do not need to buy a book for this course, but the following may be useful for
background reading. If you do buy something, the starred books are recommended
[1] Functional Analysis, W. Rudin, McGraw{Hill (1973). This book is thorough,
sophisticated and demanding.
[2] Functional Analysis, F. Riesz and B. Sz.-Nagy, Dover (1990). This is a classic
text, also much more sophisticated than the course.
[3]* Foundations of Modern Analysis, A. Friedman, Dover (1982). Cheap and
cheerful, includes a useful few sections on background.
[4]* Essential Results of Functional Analysis, R.J. Zimmer, University of Chicago
Press (1990). Lots of good problems and a useful chapter on background.
[5]* Functional Analysis in Modern Applied Mathematics, R.F. Curtain and A.J.
Pritchard, Academic Press (1977). This book is closest to the course.
[6]* Linear Analysi, B. Bollobas, Cambridge University Press (1995). This book is
excellent but makes heavy demands on the reader.
Contents
Chapter 1. Normed Linear Spaces
5
1. Linear (vector) spaces
5
2. Linear subspaces
7
3. Linear independence
7
4. Norms
7
5. Isomorphism of normed linear spaces
9
6. Products of normed spaces
9
7. Continuous maps between normed spaces
10
8. Sequences and completeness in normed spaces
12
9. Topological language
13
10. Quotient spaces
15
Chapter 2. Banach spaces
17
1. Completions
18
2. Contraction mapping theorem
19
3. Applications to dierential equations
22
4. Applications to integral equations
25
Chapter 3. Linear Transformations
29
1. Bounded operators
29
2. The space of linear operators
30
3. Banach algebras
32
4. Uniform boundedness
32
5. An application of uniform boundedness to Fourier series
34
6. Open mapping theorem
36
7. Hahn{Banach theorem
38
Chapter 4. Integration
43
1. Lebesgue measure
43
2. Product spaces and Fubini's theorem
46
Chapter 5. Hilbert spaces
47
1. Hilbert spaces
47
2. Projection theorem
50
3. Projection and self{adjoint operators
52
4. Orthonormal sets
54
5. Gram{Schmidt orthonormalization
57
Chapter 6. Fourier analysis
59
1. Fourier series of L 1 functions
59
2. Convolution in L 1
61
3
4
CONTENTS
3. Summability kernels and homogeneous Banach algebras
62
4. Fejer's kernel
64
5. Pointwise convergence
67
6. Lebesgue's Theorem
69
Appendix A.
71
1. Zorn's lemma and Hamel bases
71
2. Baire category theorem
72
CHAPTER 1
Normed Linear Spaces
A linear space is simply an abstract version of the familiar vector spaces R, R 2 ,
R 3 and so on. Recall that vector spaces have certain algebraic properties: vectors
may be added, multiplied by scalars, and vector spaces have bases and subspaces.
Linear maps between vector spaces may be described in terms of matrices. Using the
Euclidean norm or distance, vector spaces have other analytic properties (though
you may not have called them that): for example, certain functions from R to R
are continuous, dierentiable, Riemann integrable and so on.
We need to make three steps of generalization.
Bases: The rst is familiar: instead of, for example, R 3 , we shall sometimes want
to talk about an abstract three{dimensional vector space V over the eld R. This
distinction amounts to having a specic basis fe 1 ; e 2 ; e 3 g in mind, in which case
every element of V corresponds to a triple (a;b;c) = ae 1 + be 2 + ce 3 of reals { or
choosing not to think of a specic basis, in which case the elements of V are just
abstract vectors v. In the abstract language we talk about linear maps or operators
between vector spaces; after choosing a basis linear maps become matrices { though
in an innite dimensional setting it is rarely useful to think in terms of matrices.
Ground fields: The second is fairly trivial and is also familiar: the ground eld
can be any eld. We shall only be interested in R (real vector spaces) and C
(complex vector spaces). Notice that C is itself a two{dimensional vector space
over R with additional structure (multiplication). Choosing a basis f1;ig for C
over R we may identify z 2C with the vector (<(z);=(z)) 2R 2 .
Dimension: In linear algebra courses, you deal with nite dimensional vector
spaces. Such spaces (over a xed ground eld) are determined up to isomor-
phism by their dimension. We shall be mainly looking at linear spaces that are
not nite{dimensional, and several new features appear. All of these features may
be summed up in one line: the algebra of innite dimensional linear spaces is in-
timately connected to the topology. For example, linear maps between R 2 and R 2
are automatically continuous. For innite dimensional spaces, some linear maps
are not continuous.
1. Linear (vector) spaces
Definition 1.1. A linear space over a eld k is a set V equipped with maps
: V V ! V and : kV ! V with the properties
(1) xy = yx for all x;y 2 V (addition is commutative);
(2) (xy) z = x (yz) for all x;y;z 2 V (addition is associative);
(3) there is an element 0 2 V such that x 0 = 0 x = x for all x 2 V (a zero
element);
(4) for each x 2 V there is a unique element x 2 V with x (x) = 0 (additive
inverses);
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