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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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src/fa/norm/hilbert.tex
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@@ -352,7 +352,7 @@ A significant property of Hilbert spaces is that every closed subspace is comple
\phi_x: H \to \complex \quad \phi_x(y) = \dpn{y, x}{E}
\]
then the mapping $H \to H^*$ defined by $x \mapsto \phi_x$ is an isometric conjugate linear isomorphism.
then the mapping $H \to H^*$ defined by $x \mapsto \phi_x$ is an conjugate linear isometric isomorphism.
\end{theorem}
\begin{proof}
By the \hyperref[Cauchy-Schwarz inequality]{proposition:cauchy-schwarz} and definition of the norm, $x \mapsto \phi_x$ is an isometric conjugate linear map.