Markushevich basis explained

In functional analysis, a Markushevich basis (sometimes M-basis^[1]) is a biorthogonal system that is both complete and total.^[2]

Definition

Let

be Banach space. A biorthogonal system in is a Markushevich basis if $$\backslash overline\backslash \; =\; X$$and $$\backslash \_$$ separates the points of .

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

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 $$\backslash \_\backslash quad\backslash quad\backslash quad(\backslash textn=0,\backslash pm1,\backslash pm2,\backslash dots)$$ in the subspace

of continuous functions from 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 ; thus for any , there exists a sequence $$\backslash sum\_$$\to f\textBut if }, then for a fixed the coefficients must converge, and there are functions for which they do not.^[4] 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.

Notes and References

1. Book: Hušek. Miroslav. Mill. J. van. Recent Progress in General Topology II. 28 June 2014. 2002. Elsevier. 9780444509802. 182.
2. Book: Bierstedt. K.D.. Bonet. J.. Maestre. M.. J. Schmets. Recent Progress in Functional Analysis. 28 June 2014. 2001-09-20. Elsevier. 9780080515922. 4.
3. Book: Fabian . Banach Space Theory: The Basis for Linear and Nonlinear Analysis . Habala . Petr . Hájek . Petr . Montesinos Santalucía . Vicente . Zizler . Václav . 2011 . Springer . 978-1-4419-7515-7 . New York . 216–218. 10.1007/978-1-4419-7515-7 .
4. Book: Albiac . Fernando . Topics in Banach Space Theory . Kalton . Nigel J. . Springer . 2006 . 978-3-319-31557-7 . 2nd . GTM 233 . Switzerland . 2016 . 9–10 . 10.1007/978-3-319-31557-7.