The S-basis and M-basis Problems for Separable Banach Spaces

TL;DR

This paper demonstrates that every separable Banach space can be densely embedded between two Hilbert spaces and uses this to address the longstanding M-basis problem.

Contribution

It proves the existence of Hilbert space embeddings for any separable Banach space and applies this to solve the M-basis problem positively.

Findings

Existence of dense Hilbert space embeddings for all separable Banach spaces.

Application of embeddings to resolve the M-basis problem.

Improvement upon Mazur's theorem regarding Banach space embeddings.

Abstract

This note has two objectives. The first objective is show that, even if a separable Banach space does not have a Schauder basis (S-basis), there always exists Hilbert spaces $\mcH_1$ and $\mcH_2$, such that $\mcH_1$ is a continuous dense embedding in $\mcB$ and $\mcB$ is a continuous dense embedding in $\mcH_2$. This is the best possible improvement of a theorem due to Mazur (see \cite{BA} and also \cite{PE1}). The second objective is show how $\mcH_2$ allows us to provide a positive answer to the Marcinkiewicz-basis (M-basis) problem.