The Approximation Property

Abstract

Roughly speaking, the approximation problem is the question if it is true, for a given lcs E, that every operator in L (E, E) can be approximated by finite rank operators, uniformly on compact sets. If E is a Banach space, then this is equivalent to asking whether every compact operator from any Banach space with values in E can be approximated by finite rank operators in the operator norm. The problem and most of the results in this area are due to A. Grothendieck [9]. But it was P. Enflo [1] who answered the problem in the negative through a fairly involved counter-example in 1973.