Exercise 1, § 6

For every compact interval $K$ of $\mathbf{R}$, let $\mathscr{C}(K)$ be the real vector space of continuous functions on $K$, equipped with the topology of uniform convergence; denote by $\mathscr{D}(K)$ the vector space of infinitely differentiable real functions on $\mathbf{R}$ that are zero outside $K$. Equip $\mathscr{D}(K)$ with the coarsest topology for which the mappings $f \mapsto D^p f|K$ are continuous for every positive integer $p$ (D is the differentiation operator). Set $\mathscr{D} = \bigcup_K \mathscr{D}(K)$, a space that one equips with the locally convex topology that is the direct limit of the topologies of the subspaces $\mathscr{D}(K)$.

a) For every integer $p \geq 0$ and every function $f \in \mathscr{D}(K)$, set

Show that $Q_p$ is a positive quadratic form on $\mathscr{D}(K)$, that the sequence of norms $Q_p^{1/2}$ defines the topology of $\mathscr{D}(K)$, and that $\operatorname{Tr}(Q_{p+1}/Q_p)$ is finite for every $p \geq 0$. (For every real $t$, let $I_t$ be the characteristic function of the interval $[t, +\infty[$; using the formula $D^p f(t) = \int_K D^{p+1} f \cdot I_t dx$ and Bessel’s inequality, prove that $\sum_{i=1}^n Q_p(f_i) \leq l^2$ for every finite sequence $f_1, \ldots, f_n$ of functions belonging to $\mathscr{D}(K)$, orthonormal for $Q_{p+1}$; the number $l$ is the length of the interval $K$.) Deduce from this that $\mathscr{D}(K)$ is a nuclear space.

b) Prove that the space $\mathscr{D}$ is nuclear. (First establish the existence of a nonzero function in $\mathscr{D}$, and deduce from this the existence of a function $h \geq 0$ in $\mathscr{D}$ such that $\sum_{n \in \mathbf{Z}} h(x-n) = 1$ for every $x \in \mathbf{R}$. Let $K$ be a compact interval of $\mathbf{R}$ such that $h$ is zero outside of $K$; for every integer $n$, set $h_n(x) = h(x-n)$ and $K_n = K + n$. Let $V$ be a convex neighborhood of 0 in $\mathscr{D}$; there exist positive integers $p_n$ such that every function $f \in \mathscr{D}(K_n)$ with $Q_{p_n}(f) \leq 1$ belongs to $V$. Define the continuous quadratic forms $Q$ and $R$ on $\mathscr{D}$ by $Q(f) = \sum_{n \in \mathbf{Z}} 2^{2n} Q_{p_n}(f \cdot h_n)$ and $R(f) = \sum_{n \in \mathbf{Z}} 2^{3n} Q_{p_n+1}(f \cdot h_n)$; show that $V$ contains the set of $f \in \mathscr{D}$ such that $Q(f) \leq 1$ and that $\operatorname{Tr}(R/Q)$ is finite.)

c) Generalize the foregoing to infinitely differentiable functions on $\mathbf{R}^n$ with compact support.