Markushevich basis
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In functional analysis, a Markushevich basis (sometimes M-basis) is a biorthogonal system that is both ''complete'' and ''total''.


Definition

Let X be
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
. A biorthogonal system system \_ in X is a Markusevich basis if \overline\ = X and \_ separates the points of X. In a separable space, biorthogonality is not a substantial obstruction to a Markuschevich basis; any spanning set and separating functionals can be made biorthogonal. But it is an open problem whether every separable Banach space admits a Markushevich basis with \, x_\alpha\, =\, f_\alpha\, =1 for all \alpha.


Examples

Every Schauder basis of a Banach space is also a Markuschevich basis; the converse is not true in general. An example of a Markushevich basis that is not a Schauder basis is the sequence \_\quad\quad\quad(\textn=0,\pm1,\pm2,\dots) in the subspace \tilde ,1/math> of
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
s from ,1/math> 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 ,1/math>; thus for any f\in\tilde ,1/math>, there exists a sequence \sum_\to f\textBut if f=\sum_, then for a fixed n the coefficients \_N must converge, and there are functions for which they do not. 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.


References

{{mathanalysis-stub Functional analysis