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 .
