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

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 ${\text{mcH}}_1$ and ${\text{mcH}}_2$, such that ${\text{mcH}}_1$ is a continuous dense embedding in $\text{mcB}$ and $\text{mcB}$ is a continuous dense embedding in ${\text{mcH}}_2$. 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 ${\text{mcH}}_2$ allows us to provide a positive answer to the Marcinkiewicz-basis (M-basis) problem.