Fraňková-Helly selection theorem

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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.

[edit] 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 (fn)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 \mathrm{BV}([0, T]; X)

and a limit function f ∈ BV([0, T]; X) such that fn(k)(t) converges weakly in X to f(t) for every t ∈ [0, T]. That is, for every continuous linear functional λ ∈ X*,

\lambda \left( f_{n(k)}(t) \right) \to \lambda(f(t)) \mbox{ in } \mathbb{R} \mbox{ as } k \to \infty.

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

fn(t) = sin(nt).

One may ask, however, if a weaker selection theorem is true, and the Fraňková-Helly selection theorem is such a result.

[edit] 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 (fn)n∈N be a sequence in Reg([0, T]; X) satisfying the following condition: for every ε > 0, there exists some Lε > 0 so that each fn may be approximated by a un ∈ BV([0, T]; X) satisfying

\| f_{n} - u_{n} \|_{\infty} < \varepsilon

and

| u_{n}(0) | + \mathrm{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_{j=1}^{m} | u(t_{j}) - u(t_{j-1}) |

over all partitions

\Pi = \{ 0 = t_{0} < t_{1} < \dots < t_{m} = T \}

of [0, T]. Then there exists a subsequence

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

and a limit function f ∈ Reg([0, T]; X) such that fn(k)(t) converges weakly in X to f(t) for every t ∈ [0, T]. That is, for every continuous linear functional λ ∈ X*,

\lambda \left( f_{n(k)}(t) \right) \to \lambda(f(t)) \mbox{ in } \mathbb{R} \mbox{ as } k \to \infty.

[edit] References

  • Fraňková, Dana (1991). "Regulated functions". Math. Bohem. 116 (1): 20–59. ISSN 0862-7959.