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 {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*,

\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

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

\|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,m\in {\mathbf {N}}\}

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 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 \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.

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