Functional Analysis in Applied Mathematics and Engineering

Michael Pedersen

• Publisher: CRC Press
• Year: 2018
• Print ISBN: 9780849371691
• eBook ISBN: 9781351446891
• Topic: Matematica
• Subtopic: Matematica applicata

Presenting excellent material for a first course on functional analysis , Functional Analysis in Applied Mathematics and Engineering concentrates on material that will be useful to control engineers from the disciplines of electrical, mechanical, and aerospace engineering.

This text/reference discusses:

rudimentary topology

Banach's fixed point theorem with applications

L^p-spaces

density theorems for testfunctions

infinite dimensional spaces

bounded linear operators

Fourier series

open mapping and closed graph theorems

compact and differential operators

Hilbert-Schmidt operators

Volterra equations

Sobolev spaces

control theory and variational analysis

Hilbert Uniqueness Method

boundary element methods

Functional Analysis in Applied Mathematics and Engineering begins with an introduction to the important, abstract basic function spaces and operators with mathematical rigor, then studies problems in the Hilbert space setting. The author proves the spectral theorem for unbounded operators with compact inverses and goes on to present the abstract evolution semigroup theory for time dependent linear partial differential operators. This structure establishes a firm foundation for the more advanced topics discussed later in the text.

Chapter 1

Topological and Metric Spaces

In this chapter we introduce the most fundamental spaces in functional analysis, the topological spaces and the metric spaces. Since we will proceed almost immediately to the metric spaces, the topologcal spaces are defined for reference purposes only.

1.1 Some Topology

Perhaps one of the most basic features in analysis is the concept of open and closed sets. More precisely, we define a topological space S in the following way: S is a set in which we have a collection of subsets τ satisfying

$S\in \tau ,\cancel{0}\in \tau$

(1.1)

$\text{if}{A}_1,A_2,\dots A_k\in \tau ,\text{then}{\cap }_1^k{A}_j\in \tau$

(1.2)

$\text{if}{A}_j\in \tau \text{for}\text{all}j\in I,\text{then}{\cup }_{j\in I}{A}_j\in \tau ,$

(1.3)

where I is a (not necessarily finite) index set. The sets in τ are denoted open sets, and a set is defined to be closed if its complement is open. Note that sets can be both closed and open. The closure Ā of a set A is the smallest closed set containing A, and the interior of a set is the largest open set contained in it. A neighborhood of a point x in S is an open set containing x. We call (S, τ) a Hausdorff Space and τ a Hausdorff to...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• 1 Topological and Metric Spaces
• 2 Banach Spaces
• 3 Bounded Operators
• 4 Hilbert Spaces.
• 5 Operators on Hilbert Spaces
• 6 Spectral Theory
• 7 Integral Operators
• 8 Semigroups of Evolution
• 9 Sobolev Spaces
• 10 Interpolation Spaces
• 11 Linear Elliptic Operators
• 12 Regularity of Hyperbolic Mixed Problems
• 13 The Hilbert Uniqueness Method
• Exercises.
• References