In geometry, a Markushevich basis (sometimes Markushevich bases or M-basis) is a biorthogonal system that is both complete and total. It can be described by the formulation:
Let
X
be Banach space. A biorthogonal system
{
x
α
;
f
α
}
x
∈
α
in
X
is a Markusevich basis if
span
¯
{
x
α
}
=
X
and
Every Schauder basis of a Banach space is also a Markuschevich basis; the reverse is not true in general. An example of a Markushevich basis that is not a Schauder basis can be the set
{
e
2
i
π
n
t
}
n
∈
Z
in the space
C
~
[
0
,
1
]
of complex continuous functions in [0,1] whose values at 0 and 1 are equal, with the sup norm. It is an open problem whether or not every separable Banach space admits a Markushevich basis with
∥
x
α
∥
=
∥
f
α
∥
=
1
for all
α
.
