Representation of Compact Linear Operators

Abstract

Compact linear operators have a key role in functional analysis and operator theory, with a particularly important place in the study of boundary-value problems for elliptic differential equations. They have properties which are reminiscent of linear operators acting in finite-dimensional spaces, and Theorem 1.2.25 shows a Banach space Y has the approximation property (AP) if and only if given any Banach space X and any compact map T ∈ B (X, Y), T can be approximated arbitrarily closely in norm by a finite rank operator.