Fraňková–Helly selection theorem


In mathematics, the Fraňková-Helly selection theorem is a generalisation of Helly's selection theorem for functions of bounded variation to the case of regulated functions. It was proved in 1991 by the Czech mathematician Dana Fraňková.

Background

Let X be a separable Hilbert space, and let BV denote the normed vector space of all functions f : → X with finite total variation over the interval , equipped with the total variation norm. It is well known that BV satisfies the compactness theorem known as Helly's selection theorem: given any sequence of functions n∈N in BV that is uniformly bounded in the total variation norm, there exists a subsequence
and a limit function f ∈ BV such that fn converges weakly in X to f for every t ∈ . That is, for every continuous linear functional λ ∈ X*,
Consider now the Banach space Reg of all regulated functions f : → X, equipped with the supremum norm. Helly's theorem does not hold for the space Reg: a counterexample is given by the sequence
One may ask, however, if a weaker selection theorem is true, and the Fraňková-Helly selection theorem is such a result.

Statement of the Fraňková–Helly selection theorem

As before, let X be a separable Hilbert space and let Reg denote the space of regulated functions f : → X, equipped with the supremum norm. Let n∈N be a sequence in Reg satisfying the following condition: for every ε > 0, there exists some Lε > 0 so that each fn may be approximated by a un ∈ BV satisfying
and
where |-| denotes the norm in X and Var denotes the variation of u, which is defined to be the supremum
over all partitions
of . Then there exists a subsequence
and a limit function f ∈ Reg such that fn converges weakly in X to f for every t ∈ . That is, for every continuous linear functional λ ∈ X*,