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

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

and a limit function f ∈ BV([0, T]; X) such that 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*,

${\displaystyle \lambda \left(f_{n(k)}(t)\right)\to \lambda (f(t)){\mbox{ in }}X{\mbox{ as }}k\to \infty .}$

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

${\displaystyle f_{n}(t)=\sin(nt).}$

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

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

and

${\displaystyle |u_{n}(0)|+\mathrm {Var} (u_{n})\leq L_{\varepsilon },}$

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

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

over all partitions

${\displaystyle \Pi =\{0=t_{0}<t_{1}<\dots <t_{m}=T,m\in \mathbf {N} \}}$

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

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

and a limit function f ∈ Reg([0, T]; X) such that 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*,

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

References

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