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In functional analysis, a Markushevich basis (sometimes M-basis) is a biorthogonal system that is both complete and total.

Definition

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

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 ${\displaystyle \|x_{\alpha }\|=\|f_{\alpha }\|=1}$ for all ${\displaystyle \alpha }$.

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 $${\displaystyle \{e^{2i\pi nt}\}_{n\in \mathbb {Z} }\quad \quad \quad ({\text{ordered }}n=0,\pm 1,\pm 2,\dots )}$$ in the subspace ${\displaystyle {\tilde {C}}}$ of continuous functions from ${\displaystyle }$ 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 ${\displaystyle {\tilde {C}}}$; thus for any ${\displaystyle f\in {\tilde {C}}}$, there exists a sequence $${\displaystyle \sum _{|n|<N}{\alpha _{N,n}e^{2\pi int}}\to f{\text{.}}}$$But if ${\displaystyle f=\sum _{n\in \mathbb {Z} }{\alpha _{n}e^{2\pi nit}}}$, then for a fixed ${\displaystyle n}$ the coefficients ${\displaystyle \{\alpha _{N,n}\}_{N}}$ must converge, and there are functions for which they do not.

The sequence space ${\displaystyle l^{\infty }}$ admits no Markushevich basis, because it is both Grothendieck and irreflexive. But any separable space (such as ${\displaystyle l^{1}}$) has dual (resp. ${\displaystyle l^{\infty }}$) complemented in a space admitting a Markushevich basis.

References

1. Hušek, Miroslav; Mill, J. van (2002). Recent Progress in General Topology II. Elsevier. p. 182. ISBN 9780444509802. Retrieved 28 June 2014.
2. 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.
3. ^ 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.
4. 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.

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