Math 6580

One must be sure that one has enabled science to make a great advance if one
is to burden it with many new terms and require that readers follow research
that offers them so much that is strange.

A. L. Cauchy

Linear Algebra, Infinite Dimensional Spaces,

and MAPLE

PREFACE

These notes evolved in the process of teaching a beginning graduate

course in Hilbert Spaces. The first edition was simply personal, handwritten
notes prepared for lecture. A copy was typed and given to the students, for it
seemed appropriate that they should have a statement of theorems,
examples, and assignments. With each teaching of the course, the notes grew.

A text for the course has always been announced, but purchase of the

text has been optional. A text provides additional reading, alternate perspec-
tives, and a source of exercises. Whatever text was used, it was chosen with
the intent of the course in mind.

This one quarter course was designed for science and engineering

students. Often, as graduate students in the sciences and engineering
mature, they discover that the literature they are reading makes references
to function spaces, to notions of convergence, and to approximations in
unexpected norms.

The problem they face is how to learn about these ideas without

investing years of work which might carry them far from their science and
engineering studies. This course is an attempt to provide a way to understand
the ideas without the students already having the mathematical maturity
that a good undergraduate analysis course could provide. An advantage for
the instructor of this course is that the students understand that they need to
know this subject.

The course does not develop the integration theory and notion of a

measure that one should properly understand in order to discuss L

2

[0,1]. Yet,

these note suggest examples in that space. While this causes some students
to feel uneasy with their unsophisticated background, most have enough
intuition about integration that they understand the nature of the examples.
The success of the course is indicated from the fact that science and
engineering students often choose to come back the next term for a course in
real variables and in functional analysis. The course has been provocative.
Even those that do not continue with more graduate mathematics seem to
feel it serves them well and provides an opening for future conversations in
dynamics, control, and analysis.

In this most current revision of the notes, syntax for MAPLE has been

added. Many sophomores leave the calculus thinking that the computer
algebra systems are teaching tools because of the system's abilities to graph,
to take derivatives, and to solve text-book differential equations. We hope,
before they graduate, students will find that these systems really are a "way
for doing mathematics." They provide a tool for arithmetic, for solving
equations, for numerical simulations, for drawing graphs, and more. It's all in
one program! These computer algebra systems will move up the curriculum.

In choosing MAPLE, I asked for a computer algebra system which is

inexpensive for the students, which runs on a small platform, and which has
an intuitive syntax. Also, MATLAB will run Maple syntax.

The syntax given in these notes is not always the most efficient one for

writing the code. I believe that it has the advantage of being intuitive. One
hopes the student will see the code and say, "I understand that. I can do it,
too." Better yet, the student may say, "I can write better code!"

It is most important to remember that these notes are about linear

operators on Hilbert Spaces. The notes, the syntax, and the presentation
should not interfere with that subject.

The idea in these notes that has served the students best is the presen-

tation of a paradigm for a linear operator on a Hilbert Space. That paradigm
is rich enough to include all compact operators. For the student who is in the
process of studying for various exams, it provides a system for thinking of a
linear function that might have a particular property. For the student who
continues in the study of graduate mathematics, it is a proper place to step
off into a study of the spectral representations for linear operators.

To my colleagues, Neil Calkin and Eric Bussian, I say: The greatest

complement you can give a writer is to read what he has written. To my
students: These notes are better because they read them and gently
suggested changes.

I am grateful to the students through the years who have provided

suggestions, corrections, examples, and who have demanded answers for the
assignments. In time.... In time....

James V. Herod
School of Mathematics
Georgia Tech
Atlanta 30332-0160

Summer, 1997

Table of Contents

Section

Title

Page

1

A Decomposition for Matrices

1

2

Exp(tA)

5

3

Self Adjoint Transformations in Inner-Product Spaces

7

4

The Gerschgorin Circle Theorem

11

5

Convergence

14

6

Orthogonality and Closest Point Projections

18

7

Orthogonal, Nonexpansive, & Self-Adjoint Projections

23

8

Orthonormal Vectors

27

9

The Finite Dimensional Paradigm

32

10

Bounded, Linear Maps from E to I

C

35

11

Applications to Differential and Integral Equations

40

12

The Simple Paradigm for Linear Mappings from E to E

43

13

Adjoint Operators

45

14

Compact Sets

47

15

Compact Operators

50

16

The Space of Bounded Linear Transformations

53

17

The Eigenvalue Problem

57

18

Normal Operators and The More General Paradigm

61

19

Compact Operators and Orthonormal Families

64

20

The Most General Paradigm:

A Characterization of Compact Operators

67

21

The Fredholm Alternative Theorems

69

22

Closed Operators

72

23

Deficiency Index

74

24

An Application: A Problem in Control

77

26

An Application: Minimization with Constraints

??

25

A Reproducing Kernel Hilbert Space

??