Topology of ${\mathcal D}(\Omega)$ (space of test functions)

Question

I have seen two approaches to the topology of ${\mathcal D}(\Omega)$:

(i) Let $K$ be a compact subset of $\Omega$ and consider the subset ${\mathcal D}_K(\Omega)$ of test functions with support contained in $K$. This is a Fréchet space with the topology given by uniform convergence of all derivatives. Now let $K_1 \subset K_2 \subset \cdots$ be an exhaustion of $\Omega$ by compact subsets, and consider ${\mathcal D}(\Omega)$ as the inductive limit of the spaces ${\mathcal D}_{K_i}(\Omega)$.

(ii) Let ${\mathcal D}^\prime(\Omega)$ be the dual space of ${\mathcal D}(\Omega)$ in the following sense: the set of all linear functionals $f: {\mathcal D}(\Omega) \rightarrow {\mathbb R}$ such that for each compact subset $K$ of $\Omega$ there exists $C>0$ and $m \in {\mathbb N}_0$ such that $|f(\phi)|\leq C\|\phi\|_{C^{m}(K)}$ for all $\phi \in {\mathcal D}(\Omega)$ with support in $K$. Now equip ${\mathcal D}(\Omega)$ with the weak topology, i.e. the initial topology $\sigma({\mathcal D}(\Omega),{\mathcal D}^\prime(\Omega))$.

It's not completely clear to me that these topologies are the same. Can anyone help?

Answer 1

There's a general principle for proving that a topology on a vector space $E$ is not a weak topology (in the general sense, a topology of the form $\sigma(E,F)$ for some $F \subseteq E^*$).

For $\sigma(E,F)$, every neighbourhood of zero $N$ contains a linear subspace $S_N \subseteq N \subseteq E$ of finite codimension in $E$, i.e. $E/S_N$ is finite dimensional. (Easy to prove for the usual neighbourhood basis of $0$ in $\sigma(E,F)$.)

To prove topology (i) is not a weak topology, on e.g. $\mathcal{D}(\mathbb{R})$, we just find a neighbourhood of $0$ that contains no subspace of finite codimension. So take $U$ to be the set of compactly supported functions $a$ with $\sup_{x \in \mathbb{R}} |a(x)| \leq 1$ (i.e. the unit ball of the sup norm on $\mathcal{D}(\mathbb{R})$). It's easy to verify that this is a neighbourhood of 0 for the inductive limit topology (i). But its only linear subspace is $\{0\}$, which is not of finite codimension in $\mathcal{D}(\mathbb{R})$.