Added section regarding the duality of L^p spaces.

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Bokuan Li
2026-06-15 21:47:49 -04:00
parent 35efec2d90
commit dadddc4663
6 changed files with 240 additions and 7 deletions

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Let $K \in \RC$ and $\dpn{E, F}{\lambda}$ be a duality of normed vector spaces over $K$ with $E$ being separable, then
\begin{enumerate}
\item The closed unit ball $S = \bracsn{y \in F|\ \norm{y}_{F} \le 1}$ is separable with respect to the $\sigma(F, E)$-topology.
\item If the duality is norming, then there exists $\seq{y_n} \subset F$ such that for each $x \in E$, $\norm{x}_E = \sup_{n \in \natp}\dpn{x, y_n}{\lambda}$.
\item If the duality is norming, then there exists $\seq{y_n} \subset F$ such that for each $x \in E$, $\norm{x}_E = \sup_{n \in \natp}|\dpn{x, y_n}{\lambda}|$.
\end{enumerate}
\end{proposition}
\begin{proof}