Removed parts from Zhu citations.
This commit is contained in:
@@ -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.
|
||||
|
||||
Reference in New Issue
Block a user