In functional analysis, a Markushevich basis (sometimes M-basis [1]) is a biorthogonal system that is both complete and total. [2]
Let be Banach space. A biorthogonal system system in is a Markushevich basis if and separates the points of .
In a separable space, biorthogonality is not a substantial obstruction to a Markushevich 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 for all . [3]
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 in the subspace of continuous functions from 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 ; thus for any , there exists a sequence But if , then for a fixed the coefficients must converge, and there are functions for which they do not. [3] [4]
The sequence space admits no Markushevich basis, because it is both Grothendieck and irreflexive. But any separable space (such as ) has dual (resp. ) complemented in a space admitting a Markushevich basis. [3]
In functional analysis, a Markushevich basis (sometimes M-basis [1]) is a biorthogonal system that is both complete and total. [2]
Let be Banach space. A biorthogonal system system in is a Markushevich basis if and separates the points of .
In a separable space, biorthogonality is not a substantial obstruction to a Markushevich 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 for all . [3]
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 in the subspace of continuous functions from 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 ; thus for any , there exists a sequence But if , then for a fixed the coefficients must converge, and there are functions for which they do not. [3] [4]
The sequence space admits no Markushevich basis, because it is both Grothendieck and irreflexive. But any separable space (such as ) has dual (resp. ) complemented in a space admitting a Markushevich basis. [3]