Exercise 25, § 1

a) Show that every real (resp. complex) Banach space is isometric to a closed vector subspace of a Banach space of the form $\mathscr{C}(S; \mathbf{R})$ (resp. $\mathscr{C}(S; \mathbf{C})$) consisting of all real (resp. complex) continuous functions defined on a compact space S (GT, X, § 4, No. 1) (use formula (3) of IV, p. 7).

b) Deduce from $a$) that every Hausdorff locally convex space $E$ is isomorphic to a subspace of a locally convex space of the form $\mathscr{C}_c(L; \mathbf{R})$ (resp. $\mathscr{C}_c(L; \mathbf{C})$) (GT, X, § 1, No. 6). In particular, every Fréchet space is isomorphic to a closed subspace of a space $\mathscr{C}_c(L; \mathbf{R})$ (resp. $\mathscr{C}_c(L; \mathbf{C})$), where $L$ is locally compact and separable.