Linear Functional Analysis

Front Cover
Springer Science & Business Media, Jan 1, 2000 - Mathematics - 273 pages
0 Reviews
This introduction to the ideas and methods of linear functional analysis shows how familiar and useful concepts from finite-dimensional linear algebra can be extended or generalized to infinite-dimensional spaces. Aimed at advanced undergraduates in mathematics and physics, the book assumes a standard background of linear algebra, real analysis (including the theory of metric spaces), and Lebesgue integration, although an introductory chapter summarizes the requisite material.

The initial chapters develop the theory of infinite-dimensional normed spaces, in particular Hilbert spaces, after which the emphasis shifts to studying operators between such spaces. Functional analysis has applications to a vast range of areas of mathematics; the final chapters discuss the particularly important areas of integral and differential equations.

Further highlights of the second edition include:

a new chapter on the Hahn???Banach theorem and its applications to the theory of duality. This chapter also introduces the basic properties of projection operators on Banach spaces, and weak convergence of sequences in Banach spaces - topics that have applications to both linear and nonlinear functional analysis;

extended coverage of the uniform boundedness theorem;

plenty of exercises, with solutions provided at the back of the book.

  

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

Preliminaries
1
11 Linear Algebra
2
12 Metric Spaces
11
13 Lebesgue Integration
20
Normed Spaces
31
22 Finitedimensional Normed Spaces
39
23 Banach Spaces
45
Inner Product Spaces Hilbert Spaces
51
52 Normal Selfadjoint and Unitary Operators
132
53 The Spectrum of an Operator
139
54 Positive Operators and Projections
148
Compact Operators
161
62 Spectral Theory of Compact Operators
172
63 Selfadjoint Compact Operators
182
Integral and Differential Equations
191
72 Volterra Integral Equations
201

32 Orthogonality
60
33 Orthogonal Complements
65
34 Orthonormal Bases in Infinite Dimensions
72
35 Fourier Series
82
Linear Operators
87
42 The Norm of a Bounded Linear Operator
96
43 The Space BX Y and Dual Spaces
104
44 Inverses of Operators
111
Linear Operators on Hilbert Spaces
123
73 Differential Equations
203
74 Eigenvalue Problems and Greens Functions
208
Solutions to Exercises
221
Further Reading
265
References
267
Notation Index
269
Index
271
Copyright

Common terms and phrases

References to this book

Bibliographic information