Uniform Convergence of Functions

Pointwise convergence

Pointwise Convergence

Let be a set, a metric space, and a function. Consider a sequence of functions from to . We say pointwise if for all there holds

Uniform Convergence

Let be a set, a metric space, and a function. Consider a sequence of functions from to . we say uniformly on if for any there exists such that for all there holds Equivalently,

Proposition

Uniform convergence implies pointwise convergence.

Theorem

Let and be metric spaces, and let be a sequence of continuous functions such that uniformly. Then is continuous.

Theorem

Thrm Let be a sequence of continuous functions such that uniformly on for some function . Then

Proof

Def Bounded Functions Let and be metric spaces. Then

Arzela-Ascoli Theorem

Uniformly Equicontinuous

Let and be metric spaces. Let be a set of functions from to . is called uniformly equicontinuous if for any there exists such that for all for any with there holds

Proposition

If a set of functions from to , is uniformly equicontinuous, then every is uniformly continuous.

Arzela-Ascoli Theorem

Let be a compact metric space and let denote the set of continuous functions from to equipped with the uniform metric. is closed, bounded and uniformly equicontinuous iff is compact in .

Proof