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