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([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
- [math]\displaystyle{ \left( f_{n(k)} \right) \subseteq (f_{n}) \subset \mathrm{BV}([0, T]; X) }[/math]
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*,
- [math]\displaystyle{ \lambda \left( f_{n(k)}(t) \right) \to \lambda(f(t)) \mbox{ in } \mathbb{R} \mbox{ as } k \to \infty. }[/math]
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
- [math]\displaystyle{ f_{n} (t) = \sin (n t). }[/math]
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 (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
- [math]\displaystyle{ \| f_{n} - u_{n} \|_{\infty} \lt \varepsilon }[/math]
and
- [math]\displaystyle{ | u_{n}(0) | + \mathrm{Var}(u_{n}) \leq L_{\varepsilon}, }[/math]
where |-| denotes the norm in X and Var(u) denotes the variation of u, which is defined to be the supremum
- [math]\displaystyle{ \sup_{\Pi} \sum_{j=1}^{m} | u(t_{j}) - u(t_{j-1}) | }[/math]
over all partitions
- [math]\displaystyle{ \Pi = \{ 0 = t_{0} \lt t_{1} \lt \dots \lt t_{m} = T , m \in \mathbf{N} \} }[/math]
of [0, T]. Then there exists a subsequence
- [math]\displaystyle{ \left( f_{n(k)} \right) \subseteq (f_{n}) \subset \mathrm{Reg}([0, T]; X) }[/math]
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*,
- [math]\displaystyle{ \lambda \left( f_{n(k)}(t) \right) \to \lambda(f(t)) \mbox{ in } \mathbb{R} \mbox{ as } k \to \infty. }[/math]
References
 |  |
