Markushevich basis

Last updated September 18, 2024

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

Definition

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

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 $\|x_{\alpha }\|=\|f_{\alpha }\|=1$ for all $\alpha$ .^[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 $\{e^{2i\pi nt}\}_{n\in \mathbb {Z} }\quad \quad \quad ({\text{ordered }}n=0,\pm 1,\pm 2,\dots )$ in the subspace ${\tilde {C}}[0,1]$ of continuous functions from $[0,1]$ 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 ${\tilde {C}}[0,1]$ ; thus for any $f\in {\tilde {C}}[0,1]$ , there exists a sequence $\sum _{|n|<N}{\alpha _{N,n}e^{2\pi int}}\to f{\text{.}}$ But if $f=\sum _{n\in \mathbb {Z} }{\alpha _{n}e^{2\pi nit}}$ , then for a fixed $n$ the coefficients $\{\alpha _{N,n}\}_{N}$ must converge, and there are functions for which they do not.^[3]^[4]

The sequence space $l^{\infty }$ admits no Markushevich basis, because it is both Grothendieck and irreflexive. But any separable space (such as $l^{1}$ ) has dual (resp. $l^{\infty }$ ) complemented in a space admitting a Markushevich basis.^[3]

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References

↑ 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.
1 2 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.
↑ 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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[HušekMill2002-1] Hušek, Miroslav; Mill, J. van (2002). Recent Progress in General Topology II. Elsevier. p. 182. ISBN 9780444509802 . Retrieved 28 June 2014.

[BierstedtBonet2001-2] 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.

[Fabian2001-3] 1 2 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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Markushevich basis

Contents

Definition

Examples

Related Research Articles

References