Markushevich basis
Appearance
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 be Banach space. A biorthogonal system in is a Markusevich basis if
and
- separates the points in .
Every Schauder basis of a Banach space is also a Markuschevich basis; the converse is not true in general. An example of a Markushevich basis that is not a Schauder basis is the set
in the space of complex continuous functions on [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 for all .[1]
References
- ^ a b 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.