Example

— Let $\mathscr{D}'(\mathbf{R})$ be the space of distributions on $\mathbf{R}$; for $p$ a number such that $1 \leq p \leq +\infty$, let $j : L^p(\mathbf{R}) \to \mathscr{D}'(\mathbf{R})$ be the canonical injection, which is continuous (when $L^p(\mathbf{R})$ carries its normed space topology and $\mathscr{D}'(\mathbf{R})$ the strong topology). For every distribution $f \in \mathscr{D}'(\mathbf{R})$, denote the derivative of $f$ by $D(f)$; recall that $f \mapsto D(f)$ is a continuous endomorphism of $\mathscr{D}'(\mathbf{R})$. Then let E be the vector subspace of $L^p(\mathbf{R})$ formed from those $f \in L^p(\mathbf{R})$ for which $D(f) \in L^p(\mathbf{R})$, and confer on E the coarsest topology making the canonical injection $i : E \to L^p(\mathbf{R})$ and the mapping $D : E \to L^p(\mathbf{R})$ continuous ($L^p(\mathbf{R})$ carries its normed space topology). For this topology, the space E is complete. For, the image of E in $F = L^p(\mathbf{R}) \times L^p(\mathbf{R})$ by the mapping $\phi : f \mapsto (f, D(f))$ is closed, since it is the trace on $L^p(\mathbf{R}) \times L^p(\mathbf{R})$ of the image G of $\mathscr{D}'(\mathbf{R})$ in $\mathscr{D}'(\mathbf{R}) \times \mathscr{D}'(\mathbf{R})$ by the mapping

now G is the graph of $\phi_0$, therefore closed in $\mathscr{D}'(\mathbf{R}) \times \mathscr{D}'(\mathbf{R})$ (GT, I, § 8.1, cor. 2 of prop. 2), and as $\phi(E)$ is the inverse image of G by $i \times i$, which is continuous, we see that $\phi(E)$ is closed in F. \*