Compact Operators on Normed Linear Spaces
Rabindranath Sen · Anthem Press eBooks · 2013
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 E x onto a normed linear space E y is said to be compact if it maps a bounded set of ( E x ) into a compact set of ( E y ).