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

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

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

in the space ${\displaystyle {\tilde {C}}[0,1]}$ 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 ${\displaystyle \|x_{\alpha }\|=\|f_{\alpha }\|=1}$ for all ${\displaystyle \alpha }$. [1]

References

1. ^ a b 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.