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

:$left(\; f\_\{n(k)\}\; ight)\; subseteq\; (f\_\{n\})\; subset\; mathrm\{BV\}(\; [0,\; T]\; ;\; X)$

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"*,

:$lambda\; left(\; f\_\{n(k)\}(t)\; ight)\; o\; lambda(f(t))\; mbox\{\; in\; \}\; mathbb\{R\}\; mbox\{\; as\; \}\; k\; o\; infty.$

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

:$f\_\{n\}\; (t)\; =\; sin\; (n\; t).$

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

tatement 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

:

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

:

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

:$left(\; f\_\{n(k)\}\; ight)\; subseteq\; (f\_\{n\})\; subset\; mathrm\{Reg\}(\; [0,\; T]\; ;\; X)$

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"*,

:$lambda\; left(\; f\_\{n(k)\}(t)\; ight)\; o\; lambda(f(t))\; mbox\{\; in\; \}\; mathbb\{R\}\; mbox\{\; as\; \}\; k\; o\; infty.$

References

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