4.2 Continuous Linear Maps

Definition 4.2.1 (Continuous Linear Map).label Let $E,F$ be locally convex spaces over $K\in \curl{\R,\com}$ and $T\in hom(E;F)$ be a linear map then the following are equivalent:

1. (1)

   $T\in UC(E;F)$

2. (2)

   $T\in C(E;F)$

3. (3)

   $T$ is continuous at $0$.

Proposition 4.2.2.label Let $E,F$ be a locally convex spaces and $T\in hom(E;F)$ then the following are equivalent:

1. (1)

   $T$ is uniformly continuous

2. (2)

   $T$ is continuous.

3. (3)

   $T$ is continuous at $0$.

4. (4)

   For every continuous seminorm $\braks{\cd}_{F}$ on $F$, there exists a continuous seminorm $\braks{\cd}_{E}$ on $E$ such that $\braks{Tx}_{F}\leq \braks{x }_{E }$ for all $x\in E$.

Proposition 4.2.3.label Let $\curl{E_j}_{1}^{n}$ and $F$ be locally convex spaces, and $T:\prods{n}{j=1}E_{j}\to F$ be $n$-linear map, then the following are equivalent:

1. (1)

   $T$ is continuous

2. (2)

   For every continuous seminorm $\braks{\cd }_{F }$ on $F$, there exists continuous seminorms $\curl{\braks{\cd }_{j }}_{1}^{n}$ on $\curl{E_j }_{1}^{n}$ such that fro every $x\in\prods{n}{j=1}E_{j}$

   \begin{align*}\braks{Tx }_{F }\leq \prods{n}{j=1}\braks{x_j }_{E_j}\end{align*}