Abstract
In this paper, we give a comprehensive review of the classical approximation property. Then, we present some important results on modern variants, such as the weak bounded approximation property, the strong approximation property and p-approximation property. Most recent progress on E-approximation property and open problems are discussed at the end.
Highlights
In this paper, we give a comprehensive review of the classical approximation property
The first section is about classical approximation property, and some of the statements of the theorems are chosen from the beautiful review given by Casazza [1]
The second section focuses on the weak bounded approximation property and the strong approximation property introduced by Lima and Oja [2,3]
Summary
The paper was intended to give a short yet comprehensive review for the classical approximation property and variants of it. There exists a Banach space X with a basis such that X ∗ fails the approximation property and is separable. An immediate consequence of the theorem above is that if X does not have the approximation property, we can find a separable subspace Y ⊂ X so that every space Z sitting between X and Z fails the approximation property There exists a Banach space with the bounded approximation property but fails the metric approximation property. There exists a separable Banach space with the approximation property but fails the bounded approximation property. (1) Let X be a separable Banach space with a basis whose dual is separable and fails the approximation property. If a Banach space X has the bounded approximation property, does X have the metric approximation property in an equivalent norm?
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