Removed parts from Zhu citations.

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Bokuan Li
2026-07-07 11:39:54 -04:00
parent 86aba8ee4b
commit f613e65d10
11 changed files with 17 additions and 10 deletions

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@@ -136,7 +136,7 @@
\item $\norm{\Phi(x)}_B = \norm{x}_A$.
\end{enumerate}
\end{corollary}
\begin{proof}[Proof, {{\cite[II.10.7]{Zhu}}}. ]
\begin{proof}[Proof, {{\cite[10.7]{Zhu}}}. ]
(1): Since $\Phi(G(A)) \subset G(B)$, $\sigma_B(\Phi(x)) \subset \sigma_A(x)$. If $\sigma_B(\Phi(x)) \subsetneq \sigma_A(x)$, then \hyperref[Urysohn's Lemma]{lemma:urysohn} implies that there exists $C(\sigma_A(x); \complex)$ such that $f|_{\sigma_B(\Phi(x))} = 0$ but $f \ne 0$. In which case, by (7) of the \hyperref[continuous functional calculus]{definition:continuous-functional-calculus}, $\Phi(f(x)) = f(\Phi(x)) = 0$, which contradicts the fact that $\Phi$ is injective.
(2): By \autoref{corollary:c-star-unique-norm}, $\Phi$ is isometric.