Proposition 10.13.1 ([IV.4.2, SW99]).label Let $E, F$ be TVSs over $K \in RC$ and $\alg \subset \hom(E; F)$, then the following are equivalent:
- (1)
$\alg$ is uniformly equicontinuous.
- (2)
$\alg$ is equicontinuous.
- (3)
$\alg$ is equicontinuous at $0$.
- (4)
For each $V \in \cn_{F}(0)$, there exists $U \in \cn^{o}(E)$ such that $\bigcup_{T \in \alg}T(U) \subset V$.
- (5)
For each $V \in \cn_{F}(0)$, $\bigcap_{T \in \alg}T^{-1}(V) \in \cn_{E}(0)$.
Proof. (5) $\Rightarrow$ (1): Let $V \in \cn_{F}(0)$, then $U = \bigcap_{T \in \alg}T^{-1}(V) \in \cn_{E}(0)$. Thus for any $x, y \in E$ with $x - y \in U$, $Tx - Ty \in V$ for all $T \in \alg$.$\square$