Markushevich basis

In functional analysis, a Markushevich basis (sometimes M-basis) is a biorthogonal system that is both complete and total.

Definition

Let $X$ be Banach space. A biorthogonal system $\{x_{\alpha };f_{\alpha }{\}}_{x\in \alpha }$ in $X$ is a Markushevich basis if $$\overline{\text{span}}\{x_{\alpha }\}=X$$ and $$\{f_{\alpha }{\}}_{x\in \alpha }$$ separates the points of $X$.

In a separable space, biorthogonality is not a substantial obstruction to a Markushevich basis; any spanning set and separating functionals can be made biorthogonal. But it is an open problem whether every separable Banach space admits a Markushevich basis with $\Vert x_{\alpha }\Vert =\Vert f_{\alpha }\Vert =1$ for all $\alpha$.

Examples

Every Schauder basis of a Banach space is also a Markushevich basis; the converse is not true in general. An example of a Markushevich basis that is not a Schauder basis is the sequence $$\{e^{2i\pi nt}{\}}_{n\in \mathbb{Z}}(\text{ordered }n=0,\pm 1,\pm 2,\dots )$$ in the subspace $\tilde{C}[0,1]$ of continuous functions from $[0,1]$ to the complex numbers that have equal values on the boundary, under the supremum norm. The computation of a Fourier coefficient is continuous and the span dense in $\tilde{C}[0,1]$; thus for any $f\in \tilde{C}[0,1]$, there exists a sequence $$\sum _{|n|<N}{\alpha }_{N,n}{e}^{2\pi int}\to f\text{.}$$But if $f=\sum _{n\in \mathbb{Z}}{\alpha }_n{e}^{2\pi nit}$, then for a fixed $n$ the coefficients $\{{\alpha }_{N,n}{\}}_N$ must converge, and there are functions for which they do not.

The sequence space $l^{\mathrm{\infty }}$ admits no Markushevich basis, because it is both Grothendieck and irreflexive. But any separable space (such as $l^1$) has dual (resp. $l^{\mathrm{\infty }}$) complemented in a space admitting a Markushevich basis.