- Helly's selection theorem
In
mathematics , Helly's selection theorem states that asequence of functions that is locally of bounded total variation and uniformly bounded at a point has a convergentsubsequence . In other words, it is a compactness theorem for the space BVloc.It is named for the Austrian
mathematician Eduard Helly .The theorem is used in
Game theory , in particular when consideringGames on the unit square .tatement of the theorem
Let "U" be an
open subset of thereal line and let "f""n" : "U" → R, "n" ∈ N, be a sequence of functions. Suppose that
* ("f""n") has uniformly boundedtotal variation on any "W" that iscompactly embedded in "U". That is, for all sets "W" ⊆ "U" with compact closure "W̄" ⊆ "U",::sup_{n in mathbb{N left( left| f_{n} ight|_{L^{1} (W)} + left| frac{mathrm{d} f_{n{mathrm{d} t} ight|_{L^{1} (W)} ight) < + infty,:where the derivative is taken in the sense of tempered distributions;
* and ("f""n") is uniformly bounded at a point. That is, for some "t" ∈ "U", { "f""n"("t") | "n" ∈ N } ⊆ R is abounded set .Then there exists a
subsequence "f""n""k", "k" ∈ N, of "f""n" and a function "f" : "U" → R, locally ofbounded variation , such that
* "f""n""k" converges to "f" pointwise;
* and "f""n""k" converges to "f" locally in "L"1 (seelocally integrable function ), i.e., for all "W" compactly embedded in "U",::lim_{k o infty} int_{W} ig| f_{n_{k (x) - f(x) ig| , mathrm{d} x = 0;
* and, for "W" compactly embedded in "U",::left| frac{mathrm{d} f}{mathrm{d} t} ight|_{L^{1} (W)} leq liminf_{k o infty} left| frac{mathrm{d} f_{n{mathrm{d} t} ight|_{L^{1} (W)}.Generalizations
There are many generalizations and refinements of Helly's theorem. The following theorem, for BV functions taking values in
Banach space s, is due to Barbu and Precupanu:Let "X" be a reflexive, separable Banach space and let "E" be a closed, convex subset of "X". Let Δ : "X" → [0, +∞) be
positive-definite and homogeneous of degree one. Suppose that "z""n" is a uniformly bounded sequence in BV( [0, "T"] ; "X") with "z""n"("t") ∈ "E" for all "n" ∈ N and "t" ∈ [0, "T"] . Then there exists a subsequence "z""n""k" and functions "δ", "z" ∈ BV( [0, "T"] ; "X") such that
* for all "t" ∈ [0, "T"] ,::int_{ [0, t)} Delta (mathrm{d} z_{n_{k) o delta(t);
* and, for all "t" ∈ [0, "T"] ,::z_{n_{k (t) ightharpoonup z(t) in E;
* and, for all 0 ≤ "s" < "t" ≤ "T",::int_{ [s, t)} Delta(mathrm{d} z) leq delta(t) - delta(s)See also
*
Bounded variation
*Fraňková-Helly selection theorem
*Total variation References
* cite book
last = Barbu
first = V.
coauthors = Precupanu, Th.
title = Convexity and optimization in Banach spaces
series = Mathematics and its Applications (East European Series)
volume = 10
edition = Second Romanian Edition
publisher = D. Reidel Publishing Co.
location = Dordrecht
year = 1986
pages = xviii+397
isbn = 90-277-1761-3 MathSciNet|id=860772
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