Book contents
- Frontmatter
- Contents
- Introduction
- Chapter 1 Notation and Preliminary Background
- Chapter 2 Gaussian Variables. K-Convexity
- Chapter 3 Ellipsoids
- Chapter 4 Dvoretzky's Theorem
- Chapter 5 Entropy, Approximation Numbers, and Gaussian Processes
- Chapter 6 Volume Ratio
- Chapter 7 Milman's Ellipsoids
- Chapter 8 Another Proof of the QS Theorem
- Chapter 9 Volume Numbers
- Chapter 10 Weak Cotype 2
- Chapter 11 Weak Type 2
- Chapter 12 Weak Hilbert Spaces
- Chapter 13 Some Examples: The Tsirelson Spaces
- Chapter 14 Reflexivity of Weak Hilbert Spaces
- Chapter 15 Fredholm Determinants
- Final Remarks
- Bibliography
- Index
Chapter 15 - Fredholm Determinants
Published online by Cambridge University Press: 05 May 2010
- Frontmatter
- Contents
- Introduction
- Chapter 1 Notation and Preliminary Background
- Chapter 2 Gaussian Variables. K-Convexity
- Chapter 3 Ellipsoids
- Chapter 4 Dvoretzky's Theorem
- Chapter 5 Entropy, Approximation Numbers, and Gaussian Processes
- Chapter 6 Volume Ratio
- Chapter 7 Milman's Ellipsoids
- Chapter 8 Another Proof of the QS Theorem
- Chapter 9 Volume Numbers
- Chapter 10 Weak Cotype 2
- Chapter 11 Weak Type 2
- Chapter 12 Weak Hilbert Spaces
- Chapter 13 Some Examples: The Tsirelson Spaces
- Chapter 14 Reflexivity of Weak Hilbert Spaces
- Chapter 15 Fredholm Determinants
- Final Remarks
- Bibliography
- Index
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.
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- Information
- The Volume of Convex Bodies and Banach Space Geometry , pp. 223 - 234Publisher: Cambridge University PressPrint publication year: 1989