Helly's selection theorem
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In mathematics, Helly's selection theorem states that a sequence of functions that is locally of bounded total variation and uniformly bounded at a point has a convergent subsequence. In other words, it is a compactness theorem for the space BVloc.
It is named for the Austrian mathematician Eduard Helly.
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[edit] Statement of the theorem
Let U be an open subset of the real line and let fn : U → R, n ∈ N, be a sequence of functions. Suppose that
- (fn) has uniformly bounded total variation on any W that is compactly embedded in U. That is, for all sets W ⊆ U with compact closure W̄ ⊆ U,
-
- where the derivative is taken in the sense of tempered distributions;
- and (fn) is uniformly bounded at a point. That is, for some t ∈ U, { fn(t) | n ∈ N } ⊆ R is a bounded set.
Then there exists a subsequence fnk, k ∈ N, of fn and a function f : U → R, locally of bounded variation, such that
- fnk converges to f pointwise;
- and fnk converges to f locally in L1 (see locally integrable function), i.e., for all W compactly embedded in U,
- and, for W compactly embedded in U,
[edit] Generalizations
There are many generalizations and refinements of Helly's theorem. The following theorem, for BV functions taking values in Banach spaces, 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 zn is a uniformly bounded sequence in BV([0, T]; X) with zn(t) ∈ E for all n ∈ N and t ∈ [0, T]. Then there exists a subsequence znk and functions δ, z ∈ BV([0, T]; X) such that
- for all t ∈ [0, T],
- and, for all t ∈ [0, T],
- and, for all 0 ≤ s < t ≤ T,
[edit] See also
[edit] References
- Barbu, V.; Precupanu, Th. (1986). Convexity and optimization in Banach spaces, Second Romanian Edition, Mathematics and its Applications (East European Series) 10, Dordrecht: D. Reidel Publishing Co., xviii+397. ISBN 90-277-1761-3. MR860772