Chapter 15 - Fredholm Determinants

Summary

We will abbreviate approximation property to A.P. Let X be a Banach space and let 1 ≤ λ < ∞. Recall that X has the A.P. (resp. λ-A.P.) if for every ε > 0 and every compact subset K ⊂ X there is a finite rank operator u (resp. with ∥u∥ ≤ λ) such that ∥u(x) - x∥ ≤ ε for all x in K. We say that X has the bounded (resp. metric) A.P. if it has the λ-A.P. for some λ ≥ 1 (resp. for λ = 1). Finally, we say that X has the uniform A.P. (in short, U.A.P.) if there is a constant 1 ≤ λ < ∞ and a sequence of integers {k(n)∣n ≥ 1} such that for every finite dimensional subspace E ⊂ X there is an operator u : X → X with rk(u) ≤ k(dim E), ∥u∥ ≤ λ and such that u(x) = x for all x in E. The aim of this section is to prove

Theorem 15.1.Every weak Hilbert space possesses the A.P.

Corollary 15.2.Every weak Hilbert space possesses the U.A.P.

The proof of the corollary is immediate using a result of Heinrich [H] which says that a space X has the U.A.P. iff every ultrapower of X has the bounded A.P.