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 $0$.

If the above holds, then $T$ is a continuous linear map. The set $L(E;F)$ denotes the vector space of all continuous linear maps from $E$ to $F$.