The M-basis Problem for Separable Banach Spaces

TL;DR

This paper proves that every separable Banach space admits a biorthogonal system with normalized elements whose span is dense in the space, addressing the M-basis problem.

Contribution

It establishes the existence of a biorthogonal system with normalized vectors spanning any separable Banach space, solving a longstanding problem.

Findings

Existence of biorthogonal systems in separable Banach spaces

Biorthogonal systems with normalized vectors

Span of the system is dense in the space

Abstract

In this note we show that, if $\mcB$ is separable Banach space, then there is a biorthogonal system $\{x_n, x_n^*\}$ such that, the closed linear span of $\{x_n\},\bar{\left\langle {\{x_n\}}\right\rangle}=\mcB$ and $\left\| {x_n} \right\|\left\| {x_n^*} \right\| = 1$ for all $n$.