User:Asmund

Statement and first consequences

A sequence {ƒ_n}_{n∈N of continuous functions on an interval I = [a,b] is uniformly bounded if there is a number M such that}

$|f_{n}(x)|\leq M$

for every function ƒ_n belonging to the sequence, and every x ∈ [a,b]. The sequence is equicontinuous if, for every ε > 0, there exists a δ > 0 such that

$|f_{n}(x)-f_{n}(y)|<\varepsilon \,$ whenever $|x-y|<\delta \,$

for every ƒ_n belonging to the sequence. Succinctly, a sequence is equicontinuous if and only if all of its elements have the same modulus of continuity. In simplest terms, the theorem can be stated as follows:

Consider a sequence of real-valued continuous functions (ƒ_n)_n∈N defined on a closed and bounded interval [a, b] of the real line. If this sequence is uniformly bounded and equicontinuous, then there exists a subsequence (ƒ_nk) that converges uniformly.

Examples

Differentiable functions

The hypotheses of the theorem are satisfied by a uniformly bounded sequence {ƒ_n} of differentiable functions with uniformly bounded derivatives. Indeed, uniform boundedness of the derivatives implies by the mean value theorem that for all x and y,

$|f_{n}(x)-f_{n}(y)|\leq K|x-y|$

where K is the supremum of the derivatives of functions in the sequence and is independent of n. So, given ε > 0, let δ = ε/2K to verify the definition of equicontinuity of the sequence. This proves the following corollary:

• Let {ƒ_n} be a uniformly bounded sequence of real-valued differentiable functions on [a,b] such that the derivatives {ƒ′} is uniformly bounded. Then there exists a subsequence {ƒ_nk} that converges uniformly on [a,b].

If, in addition, the sequence of second derivatives is also uniformly bounded, then the derivatives also converge uniformly (up to a subsequence), and so on. Another generalization holds for continuously differentiable functions. Suppose that the functions ƒ_n are continuously differentiable with derivatives ƒ_n′. Suppose that ƒ_n′ are uniformly equicontinuous and uniformly bounded, and that the sequence ƒ_n is pointwise bounded (or just bounded at a single point). Then there is subsequence of the ƒ_n converging uniformly to a continuously differentiable function.

Lipschitz and Hölder continuous functions

The argument given above proves slightly more, specifically

• If {ƒ_n} is a uniformly bounded sequence of real valued functions on [a,b] such that each ƒ is Lipschitz continuous with the same Lipschitz constant K:

$|f_{n}(x)-f_{n}(y)|\leq K|x-y|$

for all x, y ∈ [a,b] and all ƒ_n, then there is a subsequence that converges uniformly on [a,b].

The limit function is also Lipschitz continuous with the same value K for the Lipschitz constant. A slight refinement is

• A set F of functions ƒ on [a, b] that is uniformly bounded and satisfies a Hölder condition of order α, 0 < α ≤ 1, with a fixed constant M,

$|f(x)-f(y)|\leq M\,|x-y|^{\alpha },\quad x,y\in [a,b]$

is relatively compact in C([a, b]). In particular, the unit ball of the Hölder space   C ^0, α([a, b]) is compact in C([a, b]).

This holds more generally for scalar functions on a compact metric space X satisfying a Hölder condition with respect to the metric on X.