Let $L^{2}_{loc}(\Omega)$ be a space of measurable functions on domain $\Omega$ in $\mathbb{R}^n$ with topology induced by a family of seminorms $$ p_{\alpha}(u) = \| u(x)\phi_{\alpha}(x)\|_{L^2(\Omega)} $$ where $\left\{ \phi_{\alpha} \right\} = C^{\infty}_{0}(\Omega)$. Next let $L^2_0(\Omega)$ be a space of $L^2(\Omega)$-functions with compact support with topology induced from $L^2(\Omega)$. Is it true that a linear operator $A \colon L^2_0(\Omega) \to L^2_{loc}(\Omega)$ is continuous iff for any index $\alpha$ and for any compact set $K$ there exists a constant $M>0$ such that $$ p_{\alpha}(Au) \leqslant M \|u\|_{L_2(K)} $$ for any $u \in L^2(K)$?
I looked for criteria of continuity of linear operator $T$ between locally convex spaces $(V,\{q_\beta\})$ and $(W,\{p_\alpha\})$. The necessary and sufficient condition is that for any index $\alpha$ there exist a constant $M>0$ and indexes $\beta_1,\ldots,\beta_m$ such that for any $u \in V$ we have $$ p_{\alpha}(Tu) \leqslant M ( q_{\beta_1}(u) + \ldots + q_{\beta_m}(u) ). $$ I can't apply this result in my case because here we take $u$ from the whole space $V$ and in my question $u$ belongs only to subspace $u\in L^2(K) \subset L^2(\Omega)$ (so that $M = M(K)$).