Exercise 18, § 3

Let $n\in \mathbf{N}$ and U be an open subset of $\mathbf{R}^n$. The object of this exercise is to prove that $\mathscr{D}'(U)$ is a bornological space.

If K is a compact subset of U, one endows $\mathscr{C}_K^{\infty}(U)'$ with the topology of bounded convergence. Let $\pi_K:\mathscr{D}'(U)\rightarrow \mathscr{C}_K^{\infty}(U)'$ denote the linear mapping which associates with a distribution its restriction to $\mathscr{C}_K^{\infty}(U)$; it is continuous.

Let $S\subset \mathscr{D}'(U)$ be a convex balanced bornivorous subset.

a) There exists a compact subset K of U such that Ker($\pi_K$)$\subset S$. (Otherwise, consider an increasing sequence $(K_m)_{m\in\mathbf{N}}$ of compact subsets of U whose interiors cover U and choose $f_m$ in Ker($\pi_{K_m}$)- S; then the set of the elements $mf_m$ for $m\in \mathbf{N}$ is a bounded subset not absorbed by S.)

b) The set T of distributions $f\in \mathscr{D}'(U)$ such that $f+$ Ker($\pi_K$)$\subset S$ is a convex balanced bornivorous subset.

c) The set $\pi_K(T)$ is a convex balanced bornivorous subset of $\mathscr{C}_K^{\infty}(U)'$. (Let A be a bounded subset of $\mathscr{C}_K^{\infty}(U)'$; now prove that there exists a neighbourhood W of 0 in $\mathscr{D}(U)$ such that $A\subset (W\cap \mathscr{C}_K^{\infty}(U))^{\circ}$ and deduce that A is contained in $\pi_K(W^{\circ})$, hence is absorbed by $\pi_K(T).$)

d) Deduce that S is a neighbourhood of 0, and conclude that the space $\mathscr{D}'(U)$ is bornological. (Note that $\pi_K(T)$ is a neighbourhood of 0, and observe that S contains $\pi^{-1}_K(\pi_K(T)).$)