Uniform Convergence

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.

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