Helly's selection theorem

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,

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 Hilbert 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,