Introductory Functional Analysis
With Applications to Boundary Value Problems and Finite Elements
B. Daya Reddy
Springer-Verlag, New York (1998)
ISBN 387-98307-4

This book provides an introduction to functional analysis and treats in detail its application to boundary value problems and finite elements. The book is intended for use by senior undergraduate and graduate students in mathematics, the physical sciences and engineering, who may not have been exposed to some of the conventional prerequisites (such as real analysis) for a course in functional analysis. Mature researchers wishing to learn the basic ideas of functional analysis would also find the text useful.

The text is distinguished by the fact that abstract concepts are motivated and illustrated wherever possible. Readers of this book can expect to obtain a good grounding in those aspects of functional analysis that are most relevant to a proper understanding and appreciation of the mathematical aspects of boundary value problems and the finite element method.

Table of Contents

Series Preface

Preface

Introduction

I   Linear Functional Analysis

1   Sets
1.1     The Algebra of Sets
1.2     Sets of Numbers
1.3     Rⁿ and its subsets
1.4     Relations, equivalence classes, and Zorn's Lemma
1.5     Theorem proving
1.6     Bibliographical remarks
1.7     Exercises

2   Sets of functions and Lebesgue integration
2.1     Continuous functions
2.2     Measure of sets in Rⁿ
2.3     Lebesgue integration and the space L^p(W)
2.4     Bibliographical remarks
2.5     Exercises

3   Vector spaces, normed and inner product spaces
3.1     Vector spaces and subspaces
3.2     Inner product spaces
3.3     Normed spaces
3.4     Metric spaces
3.5     Bibliographical remarks
3.6     Exercises

4   Properties of normed spaces
4.1     Sequences
4.2     Convergence of sequences of functions
4.3     Completeness
4.4     Open and closed sets, completion
4.5     Orthogonal complements in Hilbert spaces
4.6     Bibliographical remarks
4.7     Exercises

5   Linear operators
5.1     Operators
5.2     Linear operators, continuous and bounded operators
5.3     Projections
5.4     Linear functionals
5.5     Bilinear forms
5.6     Bibliographical remarks
5.7     Exercises

6   Orthonormal bases and Fourier series
6.1     Finite-dimensional spaces
6.2     Finite-dimensional inner product and normed spaces
6.3     Linear operators on finite-dimensional spaces
6.4     Fourier series in Hilbert spaces
6.5     Sturm-Liouville problems
6.6     Bibliographical remarks
6.7     Exercises

7   Distributions and Sobolev spaces
7.1     Distributions
7.2     Derivatives of distributions
7.3     The Sobolev spaces H ^m(W)
7.4     Boundary values of functions and trace theorems
7.5     The spaces H₀^m(W) and H ^-m(W)
7.6     Bibliographical remarks
7.7     Exercises

II   Elliptic Boundary Value Problems

8   Elliptic boundary value problems
8.1     Differential equations, boundary conditions and initial conditions
8.2     Linear elliptic operators
8.3     Normal boundary conditions
8.4     Green's formulas and adjoint problems
8.5     Existencs, uniqueness and regularity of solutions
8.6     Bibliograhical remarks
8.7     Exercises

9   Variational boundary value problems
9.1     A simple variational boundary value problem
9.2     Formulation of variational boundary value problems
9.3     Existence, uniqueness and regularity of solutions
9.4     Minimization of functionals
9.5     Bibliographical remarks
9.6     Exercises

10   Approximate methods of solution
10.1     The Galerkin method
10.2     Properties of Galerkin approximations
10.3     Other methods of approximation
10.4     Bibliographical remarks
10.5     Exercises

III   The Finite Element Method

11   The finite element method
11.1     The finite element method for second-order problems
11.2     One-dimensional problems
11.3     Two-dimensional problems
11.4     Fourth-order problems with Hermite families of elements
11.5     Isoparametric elements
11.6     Numerical integration
11.7     Bibliographical remarks
11.8     Exercises

12   Analysis of the finite element method
12.1     Affine families of elements
12.2     Local interpolation error estimates
12.3     Error estimates for second-order problems
12.4     Isoparametric families and numerical integration
12.5     Bibliographical remarks
12.6     Exercises

References

Solutions to Exercises

Index