Markushevich basis

In geometry, a Markushevich basis (sometimes Markushevich bases^[1] or M-basis^[2]) is a biorthogonal system that is both complete and total.^[3] 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

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

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 $\alpha$. ^[1]

References

1. Marián J. Fabian (25 May 2001). Functional Analysis and Infinite-Dimensional Geometry. Springer. pp. 188–. ISBN 978-0-387-95219-2.
2. Hušek, Miroslav; Mill, J. van (2002). Recent Progress in General Topology II. Elsevier. pp. 182–. ISBN 9780444509802. Retrieved 28 June 2014.
3. Bierstedt, K.D.; Bonet, J.; Maestre, M.; J. Schmets (2001-09-20). Recent Progress in Functional Analysis. Elsevier. pp. 4–. ISBN 9780080515922. Retrieved 28 June 2014.