Exercise 1, § 3

Let $X$ be a Hausdorff topological space, and $F$ a topological vector space (over $\mathbf{R}$ or $\mathbf{C}$). Show that on the space $\mathscr{C}(X ; F)$ of all continuous maps from $X$ into $F$, the topology of compact convergence is compatible with the vector space structure.