Markushevich basis

In functional analysis, a Markushevich basis (sometimes M-basis{{cite book|last1=Hušek|first1=Miroslav|last2=Mill|first2=J. van|title=Recent Progress in General Topology II|url=https://books.google.com/books?id=v3_PVdvJek4C&pg=PA182|accessdate=28 June 2014|year=2002|publisher=Elsevier|isbn=9780444509802|page=182}}) is a biorthogonal system that is both complete and total.{{cite book|last1=Bierstedt|first1=K.D.|last2=Bonet|first2=J.|last3=Maestre|first3=M.|author4=J. Schmets|title=Recent Progress in Functional Analysis|url=https://books.google.com/books?id=G4t3B7ZHtlgC&pg=PA4|accessdate=28 June 2014|date=2001-09-20|publisher=Elsevier|isbn=9780080515922|page=4}}

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 $\alpha$ .{{cite book |author=Fabian |first=Marián J. |url=https://link.springer.com/content/pdf/10.1007/978-1-4419-7515-7.pdf |title=Banach Space Theory: The Basis for Linear and Nonlinear Analysis |last2=Habala |first2=Petr |last3=Hájek |first3=Petr |last4=Montesinos Santalucía |first4=Vicente |last5=Zizler |first5=Václav |date=2011 |publisher=Springer |isbn=978-1-4419-7515-7 |location=New York |pages=216–218|doi=10.1007/978-1-4419-7515-7 }}

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 $\{e^{2 i \pi n t}\}_{n \isin \mathbb{Z}}\quad\quad\quad(\text{ordered }n=0,\pm1,\pm2,\dots)$ 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 $\sum_{|n| But if f=\sum_{n\in\mathbb{Z}}{\alpha_ne^{2\pi nit}}, then for a fixed n the coefficients \{\alpha_{N,n}\}_N must converge, and there are functions for which they do not. {{Cite book |last1=Albiac |first1=Fernando |url=https://link.springer.com/book/10.1007/978-3-319-31557-7 |title=Topics in Banach Space Theory |last2=Kalton |first2=Nigel J. |publisher=Springer |year=2006 |isbn=978-3-319-31557-7 |edition=2nd |series=GTM 233 |location=Switzerland |publication-date=2016 |pages=9–10 |doi=10.1007/978-3-319-31557-7}}$

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.

References

Category:Functional analysis