Markushevich basis
In functional analysis, a Markushevich basis (sometimes M-basis[1]) is a biorthogonal system that is both complete and total.[2]
Definition[edit]
Let be Banach space. A biorthogonal system system in is a Markushevich basis if
In a separable space, biorthogonality is not a substantial obstruction to a Markushevich basis; any spanning set and separating functionals can be made biorthogonal. But it is an open problem whether every separable Banach space admits a Markushevich basis with for all .[3]
Examples[edit]
Every Schauder basis of a Banach space is also a Markushevich basis; the converse is not true in general. An example of a Markushevich basis that is not a Schauder basis is the sequence
The sequence space admits no Markushevich basis, because it is both Grothendieck and irreflexive. But any separable space (such as ) has dual (resp. ) complemented in a space admitting a Markushevich basis.[3]
References[edit]
- ^ Hušek, Miroslav; Mill, J. van (2002). Recent Progress in General Topology II. Elsevier. p. 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. p. 4. ISBN 9780080515922. Retrieved 28 June 2014.
- ^ Jump up to: a b c Fabian, Marián J.; Habala, Petr; Hájek, Petr; Montesinos Santalucía, Vicente; Zizler, Václav (2011). Banach Space Theory: The Basis for Linear and Nonlinear Analysis (PDF). New York: Springer. pp. 216–218. doi:10.1007/978-1-4419-7515-7. ISBN 978-1-4419-7515-7.
- ^ Albiac, Fernando; Kalton, Nigel J. (2006). Topics in Banach Space Theory. GTM 233 (2nd ed.). Switzerland: Springer (published 2016). pp. 9–10. doi:10.1007/978-3-319-31557-7. ISBN 978-3-319-31557-7.