Markushevich basis

In geometry, a Markushevich basis (sometimes Markushevich bases or M-basis) is a biorthogonal system that is both complete and total. It can be described by the formulation:

Let $X$ be Banach space. A biorthogonal system $\{x_{\alpha };f_{\alpha }\}_{x\in \alpha }$ in $X$ is a Markusevich basis if

${\overline {\text{span}}}\{x_{\alpha }\}=X$

and

$\{f_{\alpha }\}_{x\in \alpha }$ separates the points in $X$.

Every Schauder basis of a Banach space is also a Markuschevich basis; the reverse is not true in general. An example of a Markushevich basis that is not a Schauder basis can be the set

$\{e^{2i\pi nt}\}_{n\in \mathbb {Z} }$

in the space ${\tilde {C}}[0,1]$ of complex continuous functions in [0,1] whose values at 0 and 1 are equal, with the sup norm. It is an open problem whether or not every separable Banach space admits a Markushevich basis with $\|x_{\alpha }\|=\|f_{\alpha }\|=1$ for all $\alpha$.