Proposition 8

Suppose that F is finite-dimensional. Let $s$ be an integer such that $0\leqslant s < r$ and V a relatively compact open subset of U. The linear mapping $f\mapsto f|V$ of $\mathscr{C}^r(U; F)$ into $\mathscr{C}^s(V; F)$ is compact.

Let W be the set of functions $f\in \mathscr{C}^r(U; F)$ whose partial derivatives of order $\leqslant r$ take at every point of V a value of norm less than 1. The set W is a neighbourhood of 0 in the space $\mathscr{C}^r(U; F)$. Let $\alpha$ be a multi-index such that $|\alpha |\leqslant s$. Consider the set H of functions of the form $(\partial^{\alpha}f)|V$ for $f$ in W. The set H is an equicontinuous subset of $\mathscr{C}(V; F)$ (TG, X, p. 10) by the mean value theorem (VAR, R, 2.2.3). Moreover, for every $x\in V$, the image of H under the mapping $g\mapsto g(x)$ is a bounded subset, hence a relatively compact subset, of F. By Ascoli's theorem (TG, X, p. 18, Corollary 2), the set H is relatively compact in $\mathscr{C}(V; F)$. This proves that the linear mapping $f\mapsto (\partial^{\alpha}f)|V$ of $\mathscr{C}^r(U; F)$ into $\mathscr{C}(V; F)$ is compact. The proposition then follows from Lemma 3 and Remarks 5 and 6 of III, p. 3.