Helly's selection theorem

From Wikipedia, the free encyclopedia

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 BV_loc.

It is named for the Austrian mathematician Eduard Helly.

Contents 1 Statement of the theorem 2 Generalizations 3 See also 4 References

[edit] Statement of the theorem

Let U be an open subset of the real line and let f_n : U → R, n ∈ N, be a sequence of functions. Suppose that

• (f_n) 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 (f_n) is uniformly bounded at a point. That is, for some t ∈ U, { f_n(t) | n ∈ N } ⊆ R is a bounded set.

Then there exists a subsequence f_nk, k ∈ N, of f_n and a function f : U → R, locally of bounded variation, such that

• f_nk converges to f pointwise;
• and f_nk 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 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_nk 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

• Bounded variation
• Fraňková-Helly selection theorem
• Total variation

[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