Fraňková–Helly selection theorem explained

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([0, ''T'']; X) denote the normed vector space of all functions f : [0, ''T''] → X with finite total variation over the interval [0, ''T''], equipped with the total variation norm. It is well known that BV([0, ''T'']; X) satisfies the compactness theorem known as Helly's selection theorem: given any sequence of functions (f_n)_n∈N in BV([0, ''T'']; X) that is uniformly bounded in the total variation norm, there exists a subsequence

\left( f _{n(k)} \right) \subseteq (f _{n} ) \subset BV ([ 0 , T]; X)

and a limit function f ∈ BV([0, ''T'']; X) such that f_n(k)(t) converges weakly in X to f(t) for every t ∈ [0, ''T'']. That is, for every continuous linear functional λ ∈ X*,

λ \left( f _{n(k)} (t) \right) \to λ (f(t)) in R as k \to inf ty.

Consider now the Banach space Reg([0, ''T'']; X) of all regulated functions f : [0, ''T''] → X, equipped with the supremum norm. Helly's theorem does not hold for the space Reg([0, ''T'']; X): a counterexample is given by the sequence

f _{n} (t) = \sin (n t).

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([0, ''T'']; X) denote the space of regulated functions f : [0, ''T''] → X, equipped with the supremum norm. Let (f_n)_n∈N be a sequence in Reg([0, ''T'']; X) satisfying the following condition: for every ε > 0, there exists some L_ε > 0 so that each f_n may be approximated by a u_n ∈ BV([0, ''T'']; X) satisfying

\| f _{n} - u _{n} \| _{infty} < \varepsilon

and

| u _{n} ( 0 ) | + Var (u _{n} ) \leq L _{\varepsilon} ,

where |-| denotes the norm in X and Var(u) denotes the variation of u, which is defined to be the supremum

\sup _{\Pi} \sum ^{m} _{j =1} | u(t _{j} ) - u(t _{j-1} ) |

over all partitions

\Pi = \{ 0 = t _{0} < t _{1} < ... < t _{m} = T , m \in N \}

of [0, ''T'']. Then there exists a subsequence

\left( f _{n(k)} \right) \subseteq (f _{n} ) \subset Reg ([ 0 , T]; X)

and a limit function f ∈ Reg([0, ''T'']; X) such that f_n(k)(t) converges weakly in X to f(t) for every t ∈ [0, ''T'']. That is, for every continuous linear functional &lambda; ∈ X*,

λ \left( f _{n(k)} (t) \right) \to λ (f(t)) in R as k \to inf ty.

References

• Fraňková . Dana . Regulated functions . Math. Bohem. . 116 . 1991 . 1 . 20–59 . 0862-7959 . 1100424 .