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 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 $$\|x_\alpha\|=\|f_\alpha\|=1$$ 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 $$\{e^{2 i \pi n t}\}_{n \isin \mathbb{Z}}\quad\quad\quad(\text{ordered }n=0,\pm1,\pm2,\dots)$$ 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 $$f=\sum_{n\in\mathbb{Z}}{\alpha_ne^{2\pi nit}}$$, 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 $$l^\infty$$ admits no Markushevich basis, because it is both Grothendieck and irreflexive. But any separable space (such as $$l^1$$) has dual (resp. $$l^\infty$$) complemented in a space admitting a Markushevich basis.