In
functional analysis, a Markushevich basis (sometimes M-basis
) is a
biorthogonal system that is both
''complete'' and ''total''.
Definition
Let
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
is a Markusevich basis if
and
separates the points of
.
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
for all
.
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
in the subspace