Markushevich basis

In functional analysis, a Markushevich basis (sometimes M-basis^[1]) is a biorthogonal system that is both complete and total. Completeness means that the closure of the span is all of the space.^[2]

Definition

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

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

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

A Markushevich basis is bounded iff ${\displaystyle \sup _{\alpha }\|x_{\alpha }\|\|x_{\alpha }^{*}\|<\infty }$.

A Markushevich basis ${\textstyle \left\{x_{n};x_{n}^{*}\right\}_{n=1}^{\infty }\subset X\times X^{*}}$ is strong iff ${\textstyle x\in {\overline {\operatorname {span} }}\left\{\left\langle x,x_{n}^{*}\right\rangle x_{n}\right\}_{n=1}^{\infty }}$ for all ${\textstyle x\in X}$.

Since ${\displaystyle x_{\alpha }^{*}(x_{\alpha })=1}$, we always have the lower bound ${\displaystyle \|x_{\alpha }\|\|x_{\alpha }^{*}\|\geq 1}$, and therefore ${\displaystyle \sup _{i}\|x_{i}\|\|x_{i}^{*}\|\in [1,\infty ]}$.

If ${\displaystyle \sup _{\alpha }\|x_{\alpha }\|\|x_{\alpha }^{*}\|=1}$, then we can simply scale both so that ${\displaystyle \|x_{\alpha }\|=\|x_{\alpha }^{*}\|=1}$ for all ${\displaystyle \alpha }$. 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 (^{[3]: Theorem 4.59})—Let ${\textstyle X}$ be a separable Banach space. If ${\textstyle \left\{z_{i}\right\}_{i}\subset X}$ satisfies ${\textstyle {\overline {\operatorname {span} }}\left\{z_{i}\right\}_{i}=X}$ and ${\textstyle \left\{g_{i}\right\}_{i}\subset X^{*}}$ separates points of ${\textstyle X}$, then there is a Markushevich basis ${\textstyle \left\{x_{i};x_{i}^{*}\right\}}$ of ${\textstyle X}$ such that ${\textstyle \operatorname {span} \left\{x_{i}\right\}=\operatorname {span} \left\{z_{i}\right\}}$ and ${\textstyle \operatorname {span} \left\{x_{i}^{*}\right\}=\operatorname {span} \left\{g_{i}\right\}}$.

Proof

Proof

Define ${\textstyle x_{1}=z_{1}}$ and ${\textstyle x_{1}^{*}=g_{k_{1}}/g_{k_{1}}\left(z_{1}\right)}$, where ${\textstyle k_{1}\in \mathbb {N} }$ is such that ${\textstyle g_{k_{1}}\left(z_{1}\right)\neq 0}$. Then find the smallest integer ${\textstyle h_{2}}$ such that ${\textstyle g_{h_{2}}\notin \operatorname {span} \left\{x_{1}^{*}\right\}}$. Define ${\textstyle x_{2}^{*}=g_{h_{2}}-g_{h_{2}}\left(x_{1}\right)x_{1}^{*}}$. Find an index ${\textstyle k_{2}}$ such that ${\textstyle x_{2}^{*}\left(z_{k_{2}}\right)\neq 0}$, and set ${\textstyle x_{2}=\left(z_{k_{2}}-\right.\left.x_{1}^{*}\left(z_{k_{2}}\right)x_{1}\right)/x_{2}^{*}\left(z_{k_{2}}\right)}$. Let ${\textstyle h_{3}}$ be the smallest integer such that ${\textstyle z_{h_{3}}\notin \operatorname {span} \left\{x_{1},x_{2}\right\}}$. Put ${\textstyle x_{3}=z_{h_{3}}-x_{1}^{*}\left(z_{h_{3}}\right)x_{1}-x_{2}^{*}\left(z_{h_{3}}\right)x_{2}}$ and ${\textstyle x_{3}^{*}=\left(g_{k_{3}}-g_{k_{3}}\left(x_{1}\right)x_{1}^{*}-g_{k_{3}}\left(x_{2}\right)x_{2}^{*}\right)/g_{k_{3}}\left(x_{3}\right)}$, where ${\textstyle k_{3}}$ is an index such that ${\textstyle g_{k_{3}}\left(x_{3}\right)\neq 0}$. Continue by induction. At the step ${\textstyle 2n}$ we construct ${\textstyle x_{2n}^{*}}$ first, at the step ${\textstyle 2n+1}$ we start by constructing ${\textstyle x_{2n+1}}$. It follows that ${\textstyle \operatorname {span} \left\{z_{i}\right\}_{1}^{n}\subset \operatorname {span} \left\{x_{i}\right\}_{1}^{2n}}$ and ${\textstyle \operatorname {span} \left\{g_{i}\right\}_{1}^{n}\subset \operatorname {span} \left\{x_{i}^{*}\right\}_{1}^{2n}}$. Clearly ${\textstyle x_{i}^{*}\left(x_{j}\right)=\delta _{ij}}$, ${\textstyle \operatorname {span} \left\{x_{i}\right\}\subset \operatorname {span} \left\{z_{i}\right\}}$ and ${\textstyle \operatorname {span} \left\{x_{i}^{*}\right\}\subset \operatorname {span} \left\{g_{i}\right\}}$.

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 ${\displaystyle \epsilon >0}$, there exists a Markushevich basis, such that ${\displaystyle \sup _{i}\|x_{i}\|\|x_{i}^{*}\|<1+\epsilon }$.^{[4]: Theorem 1.27} However, it is an open problem whether the lower limit is reachable. That is, whether every separable Banach space has a Markushevich basis where ${\displaystyle \|x_{i}\|\|x_{i}^{*}\|=1}$ for all ${\displaystyle i}$. That is, whether every separable Banach space has an Auerbach basis.^[3][4]

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

Theorem (^{[3]: Theorem 4.60})—Let ${\textstyle Z}$ be a closed subspace of a separable Banach space ${\textstyle X}$. Any Markushevich basis ${\textstyle \{x_{i};x_{i}^{*}\}}$ of ${\textstyle Z}$ can be extended to a Markushevich basis of ${\textstyle X}$.

Every separable Banach space admits an M-basis that is not strong.^{[4]: Proposition 1.34} Every separable Banach space admits an M-basis that is strong.^{[4]: Theorem 1.36}

Examples

Any Markushevich basis ${\displaystyle \{x_{i};x_{i}^{*}\}_{x\in i}}$ of a separable Banach space can be converted to an unbounded Markushevich basis:^[4]: 10$${\displaystyle {\begin{array}{ll}v_{2n-1}:=x_{2n-1},&v_{2n}:=x_{2n-1}+{\frac {1}{2n}}x_{2n}\\v_{2n-1}^{*}:=x_{2n-1}^{*}-2nx_{2n}^{*},&v_{2n}^{*}:=2nx_{2n}^{*}\end{array}}}$$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 $${\displaystyle \{e^{2i\pi nt}\}_{n\in \mathbb {Z} }\quad \quad \quad ({\text{ordered }}n=0,\pm 1,\pm 2,\dots )}$$ in the subspace ${\displaystyle {\tilde {C}}[0,1]}$ of continuous functions from ${\displaystyle [0,1]}$ 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 ${\displaystyle {\tilde {C}}[0,1]}$; thus for any ${\displaystyle f\in {\tilde {C}}[0,1]}$, there exists a sequence $${\displaystyle \sum _{|n|<N}{\alpha _{N,n}e^{2\pi int}}\to f{\text{.}}}$$But if ${\displaystyle f=\sum _{n\in \mathbb {Z} }{\alpha _{n}e^{2\pi nit}}}$, then for a fixed ${\displaystyle n}$ the coefficients ${\displaystyle \{\alpha _{N,n}\}_{N}}$ must converge, and there are functions for which they do not.^[3][5]

The sequence space ${\displaystyle l^{\infty }}$ admits no Markushevich basis, because it is both Grothendieck and irreflexive. But any separable space (such as ${\displaystyle l^{1}}$) has dual (resp. ${\displaystyle l^{\infty }}$) complemented in a space admitting a Markushevich basis.^[3]