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 $\{x_\alpha ; f_\alpha\}_{x \isin \alpha}$ in $X$ is a Markusevich basis if

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and

$\{ f_\alpha \}_{x \isin \alpha}$ 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

$\{e^{2 i \pi n t}\}_{n \isin \mathbb{Z}}$

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 $\alpha$.