Exercise 5, § 2

Let X be a topological space and let Y be a normed space over a non-discrete valued division ring K. Let H be a subset of $\mathfrak{F}(X; Y)$, equicontinuous at a point $x_0 \in X$, and let $k > 0$ be a real number. Then the set $H_k$ of linear combinations $\sum_i c_i u_i$ of functions $u_i \in H$, such that $\sum_i |c_i| \leq k$, is equicontinuous at the point $x_0$.