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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
Marián J. Fabian (25 May 2001). Functional Analysis and Infinite-Dimensional Geometry. Springer. pp. 188–. ISBN 978-0-387-95219-2.
Hušek, Miroslav; Mill, J. van (2002). Recent Progress in General Topology II. Elsevier. pp. 182–. ISBN 9780444509802. Retrieved 28 June 2014.
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.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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