For context:
- $\Omega$ is an open subset of $\mathbb{R}^n$.
- $\mathcal{D}(\Omega)$ is the space of test (smooth, compactly supported) functions $\Omega\to\mathbb{C}$ with a complete, unmetrizable topology $\tau$.
- $\mathcal{D}_K(\Omega)$ is the space of test functions supported on the compact set $K\subseteq \Omega$, with a Fréchet-space topology $\tau_K$ that corresponds with the subspace topology under $\tau$.
- $\mathcal{D}'(\Omega)$ is the space of distributions i.e. of continuous linear functionals $\mathcal{D}(\Omega)\to\mathbb{C}$.
All the above is as explained in chapter 1 and 6 of Rudin's Functional Analysis.
I had the following written in my personal notes: $\def\L{\Lambda} \def\DDD{\mathcal{D}} \def\W{\Omega} \def\CC{\mathbb{C}}$
Result: let $\L_1,\L_2\in\DDD'(\W)$, and $z\in\CC$, then
- $\L_1+\L_2\in\DDD'(\W)$.
- $z\L_1\in\DDD'(\W)$.
- $\L_1\cdot \L_2\in\DDD'(\W)$.
As for the proof, I merely wrote that it follows from elementary results on continuity of maps between topological vector spaces, but I could not find any result that would easily imply the result above, so I set out to prove it directly:
Fix a compact $K\subseteq \W$, and suppose $$|\L_1(\phi)| \le C_1\|\phi\|_N \ \ \ \ \text{ and } \ \ \ \ |\L_2(\phi)| \le C_2\|\phi\|_M$$ for all $\phi\in\DDD_K(\W)$. For addition, we have
$$|(\L_1+\L_2)\phi| = |\L_1(\phi)+\L_2(\phi)|\le |\L_1(\phi)|+|\L_2(\phi)| \le C_1\|\phi\|_N + C_2\|\phi\|_M \le (C_1+C_2)\|\phi\|_{\max(N,M)}.$$
For scalar multiplication, we note $$|z\L_1(\phi)| \le |z|C_1\|\phi\|_N.$$
Finally, for multiplication, we have $$|(\L_1\cdot\L_2)(\phi)| = |\L_1(\phi)\L_2(\phi)| \le C_1\|\phi\|_NC_2\|\phi\|_M \le \ldots$$
Questions:
Is the proof for addition and scalar multiplication valid?
How could one finish the proof for multiplication?