Exercise 17, § 4

Let $X$ be the locally compact space defined in GT, I, §9, Exer. 11.

a) Let $H$ be a compact subset of the locally convex space $\mathscr{C}(X; \mathbf{C})$ equipped with the topology of compact convergence; show that the functions $f \in H$ are uniformly bounded on $X$ and that there exists $c \in X$ such that, for every $x \geqslant c$, all of the functions of $H$ are constant. (Argue by contradiction; if the functions of $H$ were not uniformly bounded on $X$, there would be an increasing sequence $(x_n)$ of points of $X$ such that $\sup(|H(x_n)|) \geqslant n$, where $|H(x_n)|$ denotes the set of $|f(x_n)|$ for $f \in H$; observe that the sequence $(x_n)$ is convergent in $X$. Similarly, for every $x \in X$, let $\delta(x)$ be the supremum of the oscillations of all the functions $f \in H$ in the interval $[x, \to[$; $\delta(x)$ is finite and decreasing, therefore there exists $d \in X$ such that $\delta(x)$ is constant for $x \geqslant d$. To see that this constant $\beta$ is zero, argue by contradiction as before, on forming two increasing sequences $(s_n), (t_n)$ of points of $X$ such that $s_n \leqslant t_n \leqslant s_{n+1} \leqslant t_{n+1}$ and a sequence $(f_n)$ of functions in $H$ such that $|f_n(s_n) - f_n(t_n)| \geqslant \beta/2$.)

In particular, $\mathscr{C}(X; \mathbf{C})$ is the direct sum of $\mathscr{K}(X; \mathbf{C})$ and $\mathbf{C} \cdot 1$.

b) Show that on $X$, every measure has compact support (for every $x \in X$ let $f_x$ be the characteristic function of the interval $]\leftarrow, x]$, which is continuous and has compact support; consider on $X$ the increasing function $|\mu|(f_x)$).

c) Deduce from a) and b) that in $\mathscr{M}(X; \mathbf{C}) = \mathscr{M}^1(X; \mathbf{C}) = \mathscr{C}'(X; \mathbf{C})$, the bounded sets for the topology $\mathscr{T}$ of uniform convergence on the compact subsets of $\mathscr{C}(X; \mathbf{C})$ are the bounded sets for the ultrastrong topology (Ch. III, §1, Exer. 15). Show that $\mathscr{C}'(X; \mathbf{C})$ is not quasi-complete for the topology $\mathscr{T}$ (consider the measures $\varepsilon_x$ for $x \in X$); from this, deduce that for the topology of compact convergence, $\mathscr{C}(X; \mathbf{C})$ is neither barreled nor bornological (cf. TVS, III, §4, No. 2, Cor. 4 of Th. 1 and Exer. 18 below).

d) Show that on $\mathscr{K}(X; \mathbf{C})$ the topology $\mathscr{T}_0$ (the direct limit of the topologies of the Banach spaces $\mathscr{K}(X, K; \mathbf{C})$) is identical to the topology of uniform convergence, and that, equipped with this topology, $\mathscr{K}(X; \mathbf{C})$ is complete. (Let $V$ be a neighborhood of 0 for $\mathscr{T}_0$; for every $x \in X$, let $r_x$ be the largest number $> 0$ such that $V$ contains all of the continuous functions $f$ with support contained in $]\leftarrow, x]$ and such that $\|f\| \leqslant r_x$. Show that the infimum of $r_x$ in $X$ is $> 0$, by proving that there cannot exist an increasing sequence $(x_n)$ of points of $X$ such that $\lim_{n \to \infty} r_{x_n} = 0$.)