Characterization of Continuous Linear Transformations between Locally Convex Spaces

Theorem
Let $\Bbb F \in \set {\R, \C}$.

Let $\struct {X, \mathcal P}$ and $\struct {Y, \mathcal Q}$ be locally convex spaces over $\Bbb F$.

Let $T : X \to Y$ be a linear transformation.


 * $(1) \quad$ $T$ is everywhere continuous
 * $(2) \quad$ $T$ is continuous at ${\mathbf 0}_X$
 * $(3) \quad$ for each $q \in \mathcal Q$ there exists a real number $C_q \ge 0$ and a finite set ${\mathcal I}_q \subseteq \mathcal P$ such that:
 * $\ds \map q {T x} \le C_q \max_{p \in {\mathcal I}_q} \map p x$
 * for each $x \in X$

where we consider continuity in the standard topologies of $\struct {X, \mathcal P}$ and $\struct {Y, \mathcal Q}$.