Continuous Banach-valued functions on a compact space

Let $K$ be a compact topological space and $X$ a Banach space. The space $C(K;X)$ of continuous maps is normed by

\|u\|_{C(K;X)}=\sup_{t\in K}\|u(t)\|_X.

It is complete. A uniformly Cauchy sequence has a limit at each point by completeness of $X$ ; the same uniform Cauchy bound gives uniform convergence, and a uniform limit of continuous maps is continuous.

Closed balls

A closed ball in this space is a complete metric space. An integral equation can therefore use such a ball as the domain of a contraction, provided the integral map is shown to preserve the ball and to have Lipschitz constant less than one. Completeness alone gives neither of these two bounds.