Markushevich basis
In functional analysis, a Markushevich basis (sometimes M-basis[1]) is a biorthogonal system that is both complete and total.[2]
Definition
Let [math]\displaystyle{ X }[/math] be Banach space. A biorthogonal system system [math]\displaystyle{ \{x_\alpha ; f_\alpha\}_{x \in \alpha} }[/math] in [math]\displaystyle{ X }[/math] is a Markushevich basis if [math]\displaystyle{ \overline{\text{span}}\{x_\alpha \} = X }[/math] and [math]\displaystyle{ \{ f_\alpha \}_{x \in \alpha} }[/math] separates the points of [math]\displaystyle{ X }[/math].
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 [math]\displaystyle{ \|x_\alpha\|=\|f_\alpha\|=1 }[/math] for all [math]\displaystyle{ \alpha }[/math].[3]
Examples
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 [math]\displaystyle{ \{e^{2 i \pi n t}\}_{n \isin \mathbb{Z}}\quad\quad\quad(\text{ordered }n=0,\pm1,\pm2,\dots) }[/math] in the subspace [math]\displaystyle{ \tilde{C}[0,1] }[/math] of continuous functions from [math]\displaystyle{ [0,1] }[/math] to the complex numbers that have equal values on the boundary, under the supremum norm. The computation of a Fourier coefficient is continuous and the span dense in [math]\displaystyle{ \tilde{C}[0,1] }[/math]; thus for any [math]\displaystyle{ f\in\tilde{C}[0,1] }[/math], there exists a sequence [math]\displaystyle{ \sum_{|n|\lt N}{\alpha_{N,n}e^{2\pi int}}\to f\text{.} }[/math]But if [math]\displaystyle{ f=\sum_{n\in\mathbb{Z}}{\alpha_ne^{2\pi nit}} }[/math], then for a fixed [math]\displaystyle{ n }[/math] the coefficients [math]\displaystyle{ \{\alpha_{N,n}\}_N }[/math] must converge, and there are functions for which they do not.[3][4]
The sequence space [math]\displaystyle{ l^\infty }[/math] admits no Markushevich basis, because it is both Grothendieck and irreflexive. But any separable space (such as [math]\displaystyle{ l^1 }[/math]) has dual (resp. [math]\displaystyle{ l^\infty }[/math]) complemented in a space admitting a Markushevich basis.[3]
References
- ↑ Hušek, Miroslav; Mill, J. van (2002). Recent Progress in General Topology II. Elsevier. p. 182. ISBN 9780444509802. https://books.google.com/books?id=v3_PVdvJek4C&pg=PA182. 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. https://books.google.com/books?id=G4t3B7ZHtlgC&pg=PA4. Retrieved 28 June 2014.
- ↑ 3.0 3.1 3.2 Fabian, Marián J.; Habala, Petr; Hájek, Petr; Montesinos Santalucía, Vicente; Zizler, Václav (2011). Banach Space Theory: The Basis for Linear and Nonlinear Analysis. New York: Springer. pp. 216–218. doi:10.1007/978-1-4419-7515-7. ISBN 978-1-4419-7515-7. https://link.springer.com/content/pdf/10.1007/978-1-4419-7515-7.pdf.
- ↑ 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. https://link.springer.com/book/10.1007/978-3-319-31557-7.
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