9.2 Continuous Linear Maps
Proposition 9.2.1. Let $E, F$ be locally convex spaces and $T \in \hom(E; F)$, then the following are equivalent:
$T$ is uniformly continuous.
$T$ is continuous.
$T$ is continuous at $0$.
For every continuous seminorm $[\cdot]_{F}$ on $F$, there exists a continuous seminorm $[\cdot]_{E}$ on $E$ such that $[Tx]_{F} \le [x]_{E}$ for all $x \in E$.
Proof. $(1) \Leftrightarrow (2) \Leftrightarrow (3)$: By Definition 8.5.1.
$(2) \Rightarrow (4)$: $x \mapsto [Tx]_{F}$ is a continuous seminorm on $E$.
$(4) \Rightarrow (3)$: Let $U \in \cn_{F}(0)$ be convex, circled, and radial, then its gauge $[\cdot]_{U}$ is a continuous seminorm on $F$ by Definition 9.1.10. Thus there exists a continuous seminorm $[\cdot]_{E}$ such that $[Tx]_{U} \le [x]_{E}$. In which case, $V = \bracs{x \in E| [x]_E < 1}\in \cn_{E}(0)$ with $T(V) \subset U$. Therefore $T$ is continuous at $0$, and continuous by Definition 8.5.1.$\square$
Proposition 9.2.2. Let $\seqf{E_j}$ and $F$ be locally convex spaces, and $T: \prod_{j = 1}^{n} E_{j} \to F$ be $n$-linear map, then the following are equivalent:
$T$ is continuous.
For every continuous seminorm $[\cdot]_{F}$ on $F$, there exists continuous seminorms $\seqf{[\cdot]_j}$ on $\seqf{E_j}$, such that for every $x \in \prod_{j = 1}^{n} E_{j}$,
\[[Tx]_{F} \le \prod_{j = 1}^{n} [x_{j}]_{E_j}\]
Proof. $(1) \Rightarrow (2)$: By continuity of $T$, there exists continuous seminorms $\seqf{[\cdot]_j}$ on $\seqf{E_j}$ such that for any $x \in \prod_{j = 1}^{n} E_{j}$, $\max_{1 \le j \le n}[x_{j}]_{E_j}< 1$ implies that $[Tx]_{F} < 1$. In which case, the inequality follows from linearity.$\square$