Exercise 11, § 4

a) Let $X$ be a completely regular space and let $Y$ be a closed subspace of $X$; the mapping $\varphi : u \to u|Y$ of $C^\infty(X; \mathbf{R})$ into $C^\infty(Y; \mathbf{R})$ is a continuous linear mapping of norm $\leq 1$. If $X$ is normal, $\varphi$ is a surjective strict morphism; and if $X_Y$ denotes the quotient space of $X$ obtained by identifying all the points of $Y$ ($X_Y$ is normal by Exercise 15 of Chapter IX, § 4), then the kernel of $\varphi$ can be identified (as a normed space) with the subspace of $C^\infty(X_Y; \mathbf{R})$ consisting of functions which vanish at $\omega$, the canonical image of $Y$ in $X_Y$.

b) Suppose that $X$ is metrizable and that the frontier of $Y$ in $X$ is of countable type. Show that $\varphi : u \to u|Y$ is a strict morphism which has a right inverse (Chapter III, § 6, no. 2, Proposition 3). [Let $d$ be a metric compatible with the topology of $X$; let $(a_n)$ be a sequence of points of $Fr(Y)$, dense in $Fr(Y)$, and let $V_{n,m}$ be the set of points $x \in X$ such that $d(x, a_n) < 1/m$ and $d(x, Y) > 1/2m$; construct a family of continuous mappings $f_{n,m} : X \to [0, 1]$ such that the support of $f_{n,m}$ is contained in $V_{n,m}$ and such that $\sum_{n,m} f_{n,m}(x) = 1$ for all $x \in C Y$, the family $(f_{n,m})$ being uniformly summable in some neighbourhood of each point of $C Y$. Show then that, for every $v \in C^\infty(Y; \mathbf{R})$, the function $u$ which coincides with $v$ on $Y$ and is equal to $\sum_{n,m} v(a_n) f_{n,m}(x)$ for all $x \in C Y$ belongs to $C^\infty(X; \mathbf{R})$ and is such that $\varphi(u) = v$.] (*)