Book contents
- Frontmatter
- Preface
- Contents
- Introduction
- I Preliminaries
- II Normed Linear Spaces
- III Hilbert Space
- IV Linear Operators
- V Linear Functionals
- VI Space of Bounded Linear Functionals
- VII Closed Graph Theorem and Its Consequences
- VIII Compact Operators on Normed Linear Spaces
- IX Elements of Spectral Theory of Self-Adjoint Operators in Hilbert Spaces
- X Measure and Integration in Lp Spaces
- XI Unbounded Linear Operators
- XII The Hahn-Banach Theorem and Optimization Problems
- XIII Variational Problems
- XIV The Wavelet Analysis
- XV Dynamical Systems
- List of Symbols
- Bibliography
- Index
VIII - Compact Operators on Normed Linear Spaces
Published online by Cambridge University Press: 05 May 2013
- Frontmatter
- Preface
- Contents
- Introduction
- I Preliminaries
- II Normed Linear Spaces
- III Hilbert Space
- IV Linear Operators
- V Linear Functionals
- VI Space of Bounded Linear Functionals
- VII Closed Graph Theorem and Its Consequences
- VIII Compact Operators on Normed Linear Spaces
- IX Elements of Spectral Theory of Self-Adjoint Operators in Hilbert Spaces
- X Measure and Integration in Lp Spaces
- XI Unbounded Linear Operators
- XII The Hahn-Banach Theorem and Optimization Problems
- XIII Variational Problems
- XIV The Wavelet Analysis
- XV Dynamical Systems
- List of Symbols
- Bibliography
- Index
Summary
This chapter focusses on a natural and useful generalisation of bounded linear operators having a finite dimensional range. The concept of a compact linear operator is introduced in section 8.1. Compact linear operators often appear in applications. They play a crucial role in the theory of integral equations and in various problems of mathematical physics. The relation of compactness with weak convergence and reflexivity is highlighted. The spectral properties of a compact linear operator are studied in section 8.2. The notion of the Fredholm alternative and the relevant theorems are provided in section 8.3. Section 8.4 shows how to construct a finite rank approximations of a compact operator. A reduction of the finite rank problem to a finite dimensional problem is also given.
Compact Linear Operators
Definition: compact linear operator
A linear operator mapping a normed linear space Ex onto a normed linear space Ey is said to be compact if it maps a bounded set of (Ex) into a compact set of (Ey).
- Type
- Chapter
- Information
- A First Course in Functional AnalysisTheory and Applications, pp. 282 - 322Publisher: Anthem PressPrint publication year: 2013