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 (f_n)_n∈N in BV([0, T]; X) that is uniformly bounded in the total variation norm, there exists a subsequence

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

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*,

$\lambda (f_{n(k)}(t))\to \lambda (f(t))\text{ in }\mathbb{ℝ}\text{ as }k\to \mathrm{\infty }.$

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

$f_n(t)=\sin (nt).$

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

$\Vert f_n-u_n{\Vert }_{\mathrm{\infty }}<\epsilon$

and

$|u_n(0)|+\mathrm{V}\mathrm{a}\mathrm{r}(u_n)\le L_{\epsilon },$

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

$\underset{\mathrm{\Pi }}{\sup }\sum _{j=1}^m|u(t_j)-u(t_{j-1})|$

over all partitions

$\mathrm{\Pi }=\{0=t_0<t_1<\dots <t_m=T,m\in \mathbf{N}\}$

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

$\left(f_{n(k)}\right)\subseteq (f_n)\subset \mathrm{R}\mathrm{e}\mathrm{g}([0,T];X)$

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 λ ∈ X*,

$\lambda (f_{n(k)}(t))\to \lambda (f(t))\text{ in }\mathbb{ℝ}\text{ as }k\to \mathrm{\infty }.$

References

• Fraňková, Dana (1991). "Regulated functions". Math. Bohem. 116 (1): 20–59. ISSN 0862-7959. MR 1100424.