12.3 Linear Maps

Proposition 12.3.1.label Let $E, F$ be normed vector spaces, then the topology on $L_{b}(E; F)$ is induced by the operator norm

Theorem 12.3.2 (Linear Extension Theorem (Normed)).label Let $E$ be a normed vector space over $K \in \RC$, $F$ be a Banach space over $K$, $D \subset E$ be a dense subspace, and $T \in L(D; F)$, then:

1. (1)

   There exists an extension $\ol T \in L(E; F)$ such that $\ol T|_{D} = T$.

2. (2)

   $\normn{\ol T}_{L(E; F)}= \normn{T}_{L(D; F)}$.

3. (U)

   For any $S \in C(E; F)$ satisfying (1), $S = \ol T$.