Exercise 9, § 2

Let X be a Hausdorff topological space, and let V be a vector subspace of finite dimension n of the space $\mathscr{C}(X; \mathbf{R})$. a) Show that there exist n pair-wise disjoint, open sets $U_i$ ($1 \leq i \leq n$) in X, such that any function $f \in V$ which is identically zero in each of the $U_i$, is identically zero in X (use A, II, § 7.5, cor. 3). b) Let $x_i \in U_i$ for $1 \leq i \leq n$. Deduce from a) that there exists a constant $c > 0$ such that, for every function $f \in V$, we have