Markushevich Basis

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

Definition

Let $X$ be Banach space. A biorthogonal system $\_$ in $X$ is a Markushevich basis if $\overline\ = X$
and $\_$ 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 $\alpha$.

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 $\_\quad\quad\quad(\textn=0,\pm1,\pm2,\dots)$ in the subspace \tilde ,1/math> of continuous functions from ,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 \tilde ,1/math>; thus for any f\in\tilde ,1/math>, there exists a sequence $\sum_\to f\text$But if $f=\sum_$, then for a fixed $n$ the coefficients $\_N$ must converge, and there are functions for which they do not.

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

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Functional analysis