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 {\displaystyle i} , then it means the index set is countable. Otherwise, if the index is α {\displaystyle \alpha } , then it means the index set is not necessarily countable. Let X {\displaystyle X} be Banach space. A biorthogonal system { x α ; x α ∗ } x ∈ α {\displaystyle \{x_{\alpha };x_{\alpha }^{*}\}_{x\in \alpha }} in X {\displaystyle X} is a Markushevich basis if { x α } x ∈ α {\displaystyle \{x_{\alpha }\}_{x\in \alpha }} is complete (also called "fundamental"): span ¯ { x α } = X {\displaystyle {\overline {\text{span}}}\{x_{\alpha }\}=X} and { x α ∗ } x ∈ α {\displaystyle \{x_{\alpha }^{*}\}_{x\in \alpha }} is total: it separates the points of X {\displaystyle X} . Totality is equivalently stated as span ¯ { x α ∗ } = X ∗ {\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 span ¯ { x α ∗ } = X ∗ {\textstyle {\overline {\text{span}}}\{x_{\alpha }^{*}\}=X^{*}} under the topology induced by the operator norm on X ∗ {\displaystyle X^{*}} . A Markushevich basis is bounded iff sup α ‖ x α ‖ ‖ x α ∗ ‖ < ∞ {\displaystyle \sup _{\alpha }\|x_{\alpha }\|\|x_{\alpha }^{*}\|<\infty } . A Markushevich basis { x n ; x n ∗ } n = 1 ∞ ⊂ X × X ∗ {\textstyle \left\{x_{n};x_{n}^{*}\right\}_{n=1}^{\infty }\subset X\times X^{*}} is strong iff x ∈ span ¯ { ⟨ x , x n ∗ ⟩ x n } n = 1 ∞ {\textstyle x\in {\overline {\operatorname {span} }}\left\{\left\langle x,x_{n}^{*}\right\rangle x_{n}\right\}_{n=1}^{\infty }} for all x ∈ X {\textstyle x\in X} .
Since x α ∗ ( x α ) = 1 {\displaystyle x_{\alpha }^{*}(x_{\alpha })=1} , we always have the lower bound ‖ x α ‖ ‖ x α ∗ ‖ ≥ 1 {\displaystyle \|x_{\alpha }\|\|x_{\alpha }^{*}\|\geq 1} , and therefore sup i ‖ x i ‖ ‖ x i ∗ ‖ ∈ [ 1 , ∞ ] {\displaystyle \sup _{i}\|x_{i}\|\|x_{i}^{*}\|\in [1,\infty ]} . If sup α ‖ x α ‖ ‖ x α ∗ ‖ = 1 {\displaystyle \sup _{\alpha }\|x_{\alpha }\|\|x_{\alpha }^{*}\|=1} , then we can simply scale both so that ‖ x α ‖ = ‖ x α ∗ ‖ = 1 {\displaystyle \|x_{\alpha }\|=\|x_{\alpha }^{*}\|=1} for all α {\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:
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 {\displaystyle \epsilon >0} , there exists a Markushevich basis, such that sup i ‖ x i ‖ ‖ x i ∗ ‖ < 1 + ϵ {\displaystyle \sup _{i}\|x_{i}\|\|x_{i}^{*}\|<1+\epsilon } . 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 {\displaystyle \|x_{i}\|\|x_{i}^{*}\|=1} for all i {\displaystyle 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. Every separable Banach space admits an M-basis that is strong.
Examples Any Markushevich basis { x i ; x i ∗ } x ∈ i {\displaystyle \{x_{i};x_{i}^{*}\}_{x\in i}} of a separable Banach space can be converted to an unbounded Markushevich basis: v 2 n − 1 := x 2 n − 1 , v 2 n := x 2 n − 1 + 1 2 n x 2 n v 2 n − 1 ∗ := x 2 n − 1 ∗ − 2 n x 2 n ∗ , v 2 n ∗ := 2 n x 2 n ∗ {\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 { e 2 i π n t } n ∈ Z ( ordered n = 0 , ± 1 , ± 2 , … ) {\displaystyle \{e^{2i\pi nt}\}_{n\in \mathbb {Z} }\quad \quad \quad ({\text{ordered }}n=0,\pm 1,\pm 2,\dots )} in the subspace C ~ [ 0 , 1 ] {\displaystyle {\tilde {C}}[0,1]} of continuous functions from [ 0 , 1 ] {\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 C ~ [ 0 , 1 ] {\displaystyle {\tilde {C}}[0,1]} ; thus for any f ∈ C ~ [ 0 , 1 ] {\displaystyle f\in {\tilde {C}}[0,1]} , there exists a sequence ∑ | n | < N α N , n e 2 π i n t → f . {\displaystyle \sum _{|n|<N}{\alpha _{N,n}e^{2\pi int}}\to f{\text{.}}} But if f = ∑ n ∈ Z α n e 2 π n i t {\displaystyle f=\sum _{n\in \mathbb {Z} }{\alpha _{n}e^{2\pi nit}}} , then for a fixed n {\displaystyle n} the coefficients { α N , n } N {\displaystyle \{\alpha _{N,n}\}_{N}} must converge, and there are functions for which they do not. The sequence space l ∞ {\displaystyle l^{\infty }} admits no Markushevich basis, because it is both Grothendieck and irreflexive. But any separable space (such as l 1 {\displaystyle l^{1}} ) has dual (resp. l ∞ {\displaystyle l^{\infty }} ) complemented in a space admitting a Markushevich basis.
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