Markushevich basis

In functional analysis, a Markushevich basis (sometimes M-basis) is a biorthogonal system that is both complete and total. Completeness means that the closure of the span is all of the space.

Definition

Conventionally, if the index is $i$, then it means the index set is countable. Otherwise, if the index is $\alpha$, then it means the index set is not necessarily countable.

Let $X$ be Banach space. A biorthogonal system $\{x_{\alpha };x_{\alpha }^{\ast }{\}}_{x\in \alpha }$ in $X$ is a Markushevich basis if $\{x_{\alpha }{\}}_{x\in \alpha }$ is complete (also called "fundamental"):$$\overline{\text{span}}\{x_{\alpha }\}=X$$and $\{x_{\alpha }^{\ast }{\}}_{x\in \alpha }$ is total: it separates the points of $X$. Totality is equivalently stated as $\overline{\text{span}}\{x_{\alpha }^{\ast }\}=X^{\ast }$ where the closure is taken under the weak-star topology.

A Markushevich basis is shrinking iff we further have $\overline{\text{span}}\{x_{\alpha }^{\ast }\}=X^{\ast }$ under the topology induced by the operator norm on $X^{\ast }$.

A Markushevich basis is bounded iff $\underset{\alpha }{sup}\Vert x_{\alpha }\Vert \Vert x_{\alpha }^{\ast }\Vert <\mathrm{\infty }$.

A Markushevich basis ${\{x_n;x_n^{\ast }\}}_{n=1}^{\mathrm{\infty }}\subset X\times X^{\ast }$ is strong iff $x\in \overline{\mathrm{span}}{\left\{⟨x,x_n^{\ast }⟩x_n\right\}}_{n=1}^{\mathrm{\infty }}$ for all $x\in X$.

Since $x_{\alpha }^{\ast }(x_{\alpha })=1$, we always have the lower bound $\Vert x_{\alpha }\Vert \Vert x_{\alpha }^{\ast }\Vert \ge 1$, and therefore $\underset{i}{sup}\Vert x_i\Vert \Vert x_i^{\ast }\Vert \in [1,\mathrm{\infty }]$.

If $\underset{\alpha }{sup}\Vert x_{\alpha }\Vert \Vert x_{\alpha }^{\ast }\Vert =1$, then we can simply scale both so that $\Vert x_{\alpha }\Vert =\Vert x_{\alpha }^{\ast }\Vert =1$ for all $\alpha$. This special case of the Markushevich basis is called an Auerbach basis. Auerbach's lemma states that any finite-dimensional Banach space has an Auerbach basis.

Properties

In a separable space, Markushevich bases exist and in great abundance. Any spanning set and separating functionals can be made into a Markushevich basis by an inductive process similar to a Gram–Schmidt process:

The above construction, however, does not guarantee that the constructed basis is bounded.

It is known currently that for every separable Banach space, for any $ϵ>0$, there exists a Markushevich basis, such that $\underset{i}{sup}\Vert x_i\Vert \Vert x_i^{\ast }\Vert <1+ϵ$.Template:Reference page However, it is an open problem whether the lower limit is reachable. That is, whether every separable Banach space has a Markushevich basis where $\Vert x_i\Vert \Vert x_i^{\ast }\Vert =1$ for all $i$. That is, whether every separable Banach space has an Auerbach basis.

Similarly, any Markushevich basis of a closed subspace can be extended:

Every separable Banach space admits an M-basis that is not strong.Template:Reference page Every separable Banach space admits an M-basis that is strong.Template:Reference page

Examples

Any Markushevich basis $\{x_i;x_i^{\ast }{\}}_{x\in i}$ of a separable Banach space can be converted to an unbounded Markushevich basis:Template:Reference page$$\begin{array}{cc}{v}_{2n-1}:=x_{2n-1},& v_{2n}:=x_{2n-1}+\frac{1}{2n}{x}_{2n}\\ v_{2n-1}^{\ast }:=x_{2n-1}^{\ast }-2n{x}_{2n}^{\ast },& v_{2n}^{\ast }:=2n{x}_{2n}^{\ast }\end{array}$$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 $$\{e^{2i\pi nt}{\}}_{n\in \mathbb{Z}}(\text{ordered }n=0,\pm 1,\pm 2,\dots )$$ 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 $l^{\mathrm{\infty }}$ admits no Markushevich basis, because it is both Grothendieck and irreflexive. But any separable space (such as $l^1$) has dual (resp. $l^{\mathrm{\infty }}$) complemented in a space admitting a Markushevich basis.