Continuity of linear operators on test function with seminorm topology.

Question

I have a doubt about the following definition of continuity.

For all $k, j \in \mathbb{N}$ define the seminorm on $C^\infty_0 (\Omega)$, where $\Omega \subset \mathbb{R}^d$ is open and $K_j \subset \Omega$ is compact, as $$ P_{k,j} (f) := \sup_{x \in K_j} \{|\partial^\alpha f| : |\alpha| \leq k\}. $$

A linear map $T : C^\infty_0 (\Omega) \to C^\infty_0 (\Omega)$ is continuous if and only if, $\forall k,j \in \mathbb{N}$, there exist $m, n \in \mathbb{n}$ and $C= C(k, k)$ constant, so that

\begin{equation}\tag{1} P_{k,j} (T(f)) \leq C P_{m,n}(f) \, \, \,\forall f \in C^\infty_0(\Omega): \, \mathrm{supp} f \subset K_n. \end{equation}

My question is the following: is the characterisation of continuity correct? Don't we need something like T is continuous iff, $\forall k,j, \mathbf{n} \in \mathbb{N}$, there exist $n$ and $C$ so that (1) holds?

Indeed, a friend of mine made me notice that by requiring the existence of only one $n$, we might 'not see' part of the domain of the operator.

Thanks!

Answer 1

I think that the characterization is correct by the following result:

1. "Let $\mathcal{P}$ and $\mathcal{Q}$ are two sufficient families (sufficient means that $p(x)=0$ for every $p \in \mathcal{P}$ implies $x=0$) of semi-norms for the locally convex space $E$ and $F$. A linear application $T: E \rightarrow F$ is continuous if and only if for every $q \in \mathcal{Q}$ there exist $p_j \in \mathcal{P}$ with $j \in J$ finite and a constant $C \geq 0$ so that $q(Tx) \leq C \max_{j \in J} p_j(x)$."

Therefore the application $T : C^\infty_0 (\Omega) \to C^\infty_0 (\Omega)$ is continuous considering the restrictions $C_{0,K_j}^\infty(\Omega) \subset C_0^{\infty}(\Omega)$ for each $K_j \subset \Omega$.