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]

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 ${\textstyle {\overline {\text{span}}}\{x_{\alpha }^{*}\}=X^{*}}$ 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 $X^{*}$ .

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

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

If $\sup _{\alpha }\|x_{\alpha }\|\|x_{\alpha }^{*}\|=1$ , then we can simply scale both so that $\|x_{\alpha }\|=\|x_{\alpha }^{*}\|=1$ 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 (^[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 $\epsilon >0$ , there exists a Markushevich basis, such that $\sup _{i}\|x_{i}\|\|x_{i}^{*}\|<1+\epsilon$ .^[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 $\|x_{i}\|\|x_{i}^{*}\|=1$ for all $i$ . 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 $\{x_{i};x_{i}^{*}\}_{x\in i}$ of a separable Banach space can be converted to an unbounded Markushevich basis:^[4]^: 10 ${\begin{array}{ll}v_{2n-1}:=x_{2n-1},&v_{2n}:=x_{2n-1}+{\frac {1}{2n}}x_{2n}\\v_{2n-1}^{*}:=x_{2n-1}^{*}-2nx_{2n}^{*},&v_{2n}^{*}:=2nx_{2n}^{*}\end{array}}$ 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 $\{e^{2i\pi nt}\}_{n\in \mathbb {Z} }\quad \quad \quad ({\text{ordered }}n=0,\pm 1,\pm 2,\dots )$ 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.^[3]^[5]

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.^[3]

Markushevich basis

Properties

Examples

References

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