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 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 $\alpha$.

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.

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.