Definition 11.1.14 (Associated Normed Space).label Let $E$ be a separated locally convex space and $A \subset E$ be convex and circled. Let $E_{0} = \bigcup_{n \in \natp}nA$, $\rho_{0}: E_{0} \to [0, \infty)$ be the gauge of $A$, and $(E_{A}, \rho_{A})$ be the quotient of $E_{0}$ by $\bracs{\phi = 0}$, equipped with the quotient norm of $\rho_{0}$, then
- (1)
$(E_{A}, \rho_{A})$ is a normed space.
If $A$ is radial, then $E_{0} = E$ and the map $\pi_{A}: E \to E_{A}$ is the canonical projection, and
- (2)
If $A \in \cn_{E}(0)$, then $\pi_{A} \in L(E; E_{A})$.
If $(E_{0}, \rho_{0})$ is separated, then $(E_{0}, \rho_{0}) = (E_{A}, \rho_{A})$, and the map $\iota_{A}: E_{A} \to E$ is the canonical inclusion. In particular, if $A$ is bounded, then
- (3)
$(E_{0}, \rho_{0})$ is separated.
- (4)
$\iota_{A} \in L(E_{A}; E)$.
The space $(E_{A}, \rho_{A})$ is the normed space associated with $A$.
Proof. (3): Let $x \in E_{0} \setminus \bracs{0}$. Since $E$ is separated, there exists $U \in \cn_{E}(0)$ such that $x \not\in U$. As $A$ is bounded, there exists $\lambda > 0$ such that $\lambda U \supset A$. In which case, $x \not\in \lambda^{-1}A$, and $E_{0}$ is separated.
(4): Let $U \in \cn_{E}(0)$, then there exists $\lambda > 0$ such that $\lambda U \supset A$, so $\iota_{A}^{-1}(U) \supset \lambda^{-1}A \in \cn_{E_A}(0)$, and $\iota_{A}$ is continuous by Proposition 11.2.1.$\square$
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