Markushevich basis
In functional analysis, a Markushevich basis is a biorthogonal system that is both complete and total. Completeness means that the closure of the span is all of the space.
Definition
Conventionally, if the index is, then it means the index set is countable. Otherwise, if the index is, then it means the index set is not necessarily countable.Let be Banach space. A biorthogonal system in is a Markushevich basis if is complete :and is total: it separates the points of. Totality is equivalently stated as where the closure is taken under the weak-star topology.
A Markushevich basis is shrinking iff we further have under the topology induced by the operator norm on.
A Markushevich basis is bounded iff.
A Markushevich basis is strong iff for all.
Since, we always have the lower bound, and therefore.
If, then we can simply scale both so that for all. This special case of the Markushevich basis is called an Auerbach basis. Auerbach's lemma states that any finite-dimensional Banach space has an Auerbach basis.
Properties
In a separable space, Markushevich bases exist and in great abundance. Any spanning set and separating functionals can be made into a Markushevich basis by an inductive process similar to a Gram–Schmidt process:The above construction, however, does not guarantee that the constructed basis is bounded.
It is known currently that for every separable Banach space, for any, there exists a Markushevich basis, such that. However, it is an open problem whether the lower limit is reachable. That is, whether every separable Banach space has a Markushevich basis where for all. That is, whether every separable Banach space has an Auerbach basis.
Similarly, any Markushevich basis of a closed subspace can be extended:
Every separable Banach space admits an M-basis that is not strong. Every separable Banach space admits an M-basis that is strong.
Examples
Any Markushevich basis of a separable Banach space can be converted to an unbounded Markushevich basis: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 has dual complemented in a space admitting a Markushevich basis.