§ Equicontinuity, Arzela Ascoli
- A sequence/family of functions are said to be equicontinuous if they vary equally in a given nbhd
- Necessary for Arzela Ascoli
- A subset of , space of continuous functions on a compact Hausdorff space is compact iff if it is closed, bounded, and equicontinuous.
- Corollay: a sequence of is uniformly convergent iff it is equicontinuous.
- Thus, an equicontinious family converges pointwise, and moreover, since it is uniform convergence, the limit will also be continuous.
§ Uniform boundedness Principle
The uniform boundedness principle states that a pointwise bounded family of continuous linear operators between Banach spaces is equicontinuous.
§ Equicontinuity of metric spaces
- Let be metric space.
- The family is equicontinuous at a point if for all , there is a such that for all and all .
- Thus, for a given , there is a uniform choice of that works for all functions .
- It's like uniform continutity, except the uniformity is enforced acrorss the functions , not on the points on the domain.
- The family is pointwise equicontinuous iff it is equicontinuous at each point .