§ 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 $C(X)$, space of continuous functions on a compact Hausdorff space $X$ is compact iff if it is closed, bounded, and equicontinuous.
• Corollay: a sequence of $C(X)$ 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 $X, Y$ be metric space.
• The family $F$ is equicontinuous at a point $p \in X$ if for all $\epsilon > 0$, there is a $\delta > 0$ such that $d(f(p), f(x)) < \epsilon$ for all $f \in F$and all $d(p, x) < \delta$.
• Thus, for a given $\epsilon$, there is a uniform choice of $\delta$ that works for all functions .
• It's like uniform continutity, except the uniformity is enforced acrorss the functions $f_i$, not on the points on the domain.
• The family $F$ is pointwise equicontinuous iff it is equicontinuous at each point $p$.