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

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

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

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

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

If $\underset{\alpha }{\sup }\Vert x_{\alpha }\Vert \Vert x_{\alpha }^{*}\Vert =1$, then we can simply scale both so that $\Vert x_{\alpha }\Vert =\Vert x_{\alpha }^{*}\Vert =1$ 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:

Theorem () — Let $X$ be a separable Banach space. If ${\left\{z_i\right\}}_i\subset X$ satisfies $\overline{\mathrm{span}}{\left\{z_i\right\}}_i=X$ and ${\left\{g_i\right\}}_i\subset X^{*}$ separates points of $X$, then there is a Markushevich basis $\{x_i;x_i^{*}\}$ of $X$ such that $\mathrm{span}\left\{x_i\right\}=\mathrm{span}\left\{z_i\right\}$ and $\mathrm{span}\left\{x_i^{*}\right\}=\mathrm{span}\left\{g_i\right\}$.

Proof

Proof

Define $x_1=z_1$ and $x_1^{*}=g_{k_1}/g_{k_1}\left(z_1\right)$, where $k_1\in \mathbb{\mathbb{N}}$ is such that $g_{k_1}\left(z_1\right)\ne 0$. Then find the smallest integer $h_2$ such that $g_{h_2}\notin \mathrm{span}\left\{x_1^{*}\right\}$. Define $x_2^{*}=g_{h_2}-g_{h_2}\left(x_1\right)x_1^{*}$. Find an index $k_2$ such that $x_2^{*}\left(z_{k_2}\right)\ne 0$, and set $x_2=(z_{k_2}-x_1^{*}\left(z_{k_2}\right)x_1)/x_2^{*}\left(z_{k_2}\right)$. Let $h_3$ be the smallest integer such that $z_{h_3}\notin \mathrm{span}\{x_1,x_2\}$. Put $x_3=z_{h_3}-x_1^{*}\left(z_{h_3}\right)x_1-x_2^{*}\left(z_{h_3}\right)x_2$ and $x_3^{*}=(g_{k_3}-g_{k_3}\left(x_1\right)x_1^{*}-g_{k_3}\left(x_2\right)x_2^{*})/g_{k_3}\left(x_3\right)$, where $k_3$ is an index such that $g_{k_3}\left(x_3\right)\ne 0$. Continue by induction. At the step $2n$ we construct $x_{2n}^{*}$ first, at the step $2n+1$ we start by constructing $x_{2n+1}$. It follows that $\mathrm{span}{\left\{z_i\right\}}_1^n\subset \mathrm{span}{\left\{x_i\right\}}_1^{2n}$ and $\mathrm{span}{\left\{g_i\right\}}_1^n\subset \mathrm{span}{\left\{x_i^{*}\right\}}_1^{2n}$. Clearly $x_i^{*}\left(x_j\right)={\delta }_{ij}$, $\mathrm{span}\left\{x_i\right\}\subset \mathrm{span}\left\{z_i\right\}$ and $\mathrm{span}\left\{x_i^{*}\right\}\subset \mathrm{span}\left\{g_i\right\}$.

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 $\underset{i}{\sup }\Vert x_i\Vert \Vert x_i^{*}\Vert <1+ϵ$. However, it is an open problem whether the lower limit is reachable. That is, whether every separable Banach space has a Markushevich basis where $\Vert x_i\Vert \Vert x_i^{*}\Vert =1$ 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:

Theorem () — Let $Z$ be a closed subspace of a separable Banach space $X$. Any Markushevich basis $\{x_i;x_i^{*}\}$ of $Z$ can be extended to a Markushevich basis of $X$.

Every separable Banach space admits an M-basis that is not strong. Every separable Banach space admits an M-basis that is strong.

Examples

Any Markushevich basis $\{x_i;x_i^{*}{\}}_{x\in i}$ of a separable Banach space can be converted to an unbounded Markushevich basis:$$\begin{array}{ll}{v}_{2n-1}:=x_{2n-1},& v_{2n}:=x_{2n-1}+\frac{1}{2n}{x}_{2n}\\ v_{2n-1}^{*}:=x_{2n-1}^{*}-2n{x}_{2n}^{*},& v_{2n}^{*}:=2n{x}_{2n}^{*}\end{array}$$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 $$\{e^{2i\pi nt}{\}}_{n\in \mathbb{\mathbb{Z}}}(\text{ordered }n=0,\pm 1,\pm 2,\dots )$$ 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 $f=\sum _{n\in \mathbb{\mathbb{Z}}}{\alpha }_n{e}^{2\pi nit}$, 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.