Some Banach spaces are almost Hilbert

TL;DR

This paper demonstrates that uniformly convex Banach spaces with a Schauder basis can be closely approximated by Hilbert spaces, enabling the extension of Hilbert space operator theories, including Schatten classes, to these Banach spaces.

Contribution

It introduces a Hilbert space representation for the dual of uniformly convex Banach spaces with a Schauder basis, allowing classical operator theory to be extended to these spaces.

Findings

The dual space of a uniformly convex Banach space has a Hilbert space representation.

Operators on such Banach spaces have well-defined adjoints, forming a *-algebra.

Schatten class theory extends naturally to these Banach spaces.

Abstract

The purpose of this note is to show that, if $\mcB$ is a uniformly convex Banach, then the dual space $\mcB'$ has a "Hilbert space representation" (defined in the paper), that makes $\mcB$ much closer to a Hilbert space then previously suspected. As an application, we prove that, if $\mcB$ also has a Schauder basis (S-basis), then for each $A \in \C[\mcB]$ (the closed and densely defined linear operators), there exists a closed densely defined linear operator $A^* \in \C[\mcB]$ that has all the expected properties of an adjoint. Thus for example, the bounded linear operators, $L[\mcB]$, is a $^*$algebra. This result allows us to give a natural definition to the Schatten class of operators on a uniformly convex Banach space with a S-basis. In particular, every theorem that is true for the Schatten class on a Hilbert space, is also true on such a space. The main tool we use is a special version of a result due to Kuelbs \cite{K}, which shows that every uniformly convex Banach space with a S-basis can be densely and continuously embedded into a Hilbert space which is unique up to a change of basis.