First typo fix of spectral II.

src/op/vn/fc.tex

@@ -27,7 +27,7 @@
     \end{enumerate}
 
     Let $\mathscr{E} \subset M_R(X; \complex)$ be the closed subspace generated by $\bracsn{E_{x, y}|x, y \in H}$, then
-    \begin{enumerate}[start=1]
+    \begin{enumerate}[start=2]
         \item For any $\mu \in \mathscr{E}$ and $\nu \in M_R(X; \complex)$ with $\nu \ll \mu$, $\nu \in \mathscr{E}$ as well. 
         \item Let 
         \[
@@ -223,7 +223,7 @@
 
     because $\ol{A\xi_i} \perp \ol{A\xi_j}$, so $E_{x, y} = 0$. In particular, $\seqi{\mu}$ is a mutually singular family of measures, and $[l^1(I); L^1(\mu_i; \complex)]$ may be identified as a subspace of $M_R(\Omega(A); \complex)$. 
 
-    (2): For any $x, y \in H$, $\norm{E_{x, y}}_{\text{var}} \le \norm{x}_H \norm{y}_H$ by (1) of \autoref{lemma:spectral-measure-properties}. By (1), $\bracsn{E_{x, y}|x, y \in \ol{A\xi_i}} \subset [l^1(I); L^1(\mu_i; \complex)]$ for all $i \in I$. For each $i \in I$, let $P_i \in B(H)$ be the orthogonal projection of $H$ onto $\ol{A\xi_i}$, then as $\seqi{\xi}$ is maximal, for any $x = \sum_{i \in I}P_ix$ for all $x \in H$. Thus for any $x, y \in H$, 
+    (2): For any $x, y \in H$, $\norm{E_{x, y}}_{\text{var}} \le \norm{x}_H \norm{y}_H$ by (1) of \autoref{lemma:spectral-measure-properties}. By (1), $\bracsn{E_{x, y}|x, y \in \ol{A\xi_i}} \subset [l^1(I); L^1(\mu_i; \complex)]$ for all $i \in I$. For each $i \in I$, let $P_i \in B(H)$ be the orthogonal projection of $H$ onto $\ol{A\xi_i}$, then as $\seqi{\xi}$ is maximal, $x = \sum_{i \in I}P_ix$ for all $x \in H$. Thus for any $x, y \in H$, 
     \[
         E_{x, y} = \sum_{i, j \in I}E_{P_ix, P_jy} =  \sum_{i \in I}E_{P_ix, P_iy} \in [l^1(I); L^1(\mu_i; \complex)]
     \]
@@ -232,7 +232,7 @@
 
     On the other hand, for each $i \in I$, since $\mu_i$ is a Radon measure, $C(\Omega(A); \complex)$ is dense in $L^1(\mu_i; \complex)$ by \autoref{proposition:radon-cc-dense}. As
     \[
-    \mathscr{E} \supset \bracsn{E_{x, y}|x, y \in I} \supset \bracsn{fdE_{\xi_i, \xi_i}|f \in C(\Omega(A); \complex)}  = \bracsn{fd\mu_i|f \in C(\Omega(A); \complex)} 
+    \mathscr{E} \supset \bracsn{E_{x, y}|x, y \in \ol{A\xi_i}} \supset \bracsn{fdE_{\xi_i, \xi_i}|f \in C(\Omega(A); \complex)}  = \bracsn{fd\mu_i|f \in C(\Omega(A); \complex)} 
     \]
 
     and $\mathscr{E} \subset M_R(\Omega(A); \complex)$ is closed, $\mathscr{E} \supset \bracsn{f d\mu_i|f \in L^1(\mu_i; \complex)}$. 
@@ -241,7 +241,7 @@
     
     (3): Fix $i \in I$, then for any $S, T \in A$ with $S\xi_i = T\xi_i$,
     \[
-    \Gamma_AS dE_{\xi_i, \xi_i} = E_{\Gamma_A S\xi_i, \xi_i} = E_{\Gamma_A T\xi_i, \xi_i} = \Gamma_A T dE_{\xi_i, \xi_i}
+    \Gamma_AS dE_{\xi_i, \xi_i} = E_{S\xi_i, \xi_i} = E_{T\xi_i, \xi_i} = \Gamma_A T dE_{\xi_i, \xi_i}
     \]
 
     so $\Gamma_A S = \Gamma_A T$ $\mu_i$-almost everywhere. Thus the mapping
@@ -251,7 +251,7 @@
 
     is well-defined. Moreover, for any $S, T \in A$, 
     \[
-    \dpn{S\xi_i, T\xi_i}{H} = \int \Gamma_AS \cdot \ol{\Gamma_A T} dE_{\xi_i, \xi_i} = \dpn{\Gamma S, \Gamma T}{L^2(\mu_i; \complex)}
+    \dpn{S\xi_i, T\xi_i}{H} = \int \Gamma_AS \cdot \ol{\Gamma_A T} dE_{\xi_i, \xi_i} = \dpn{\Gamma_A S, \Gamma_A T}{L^2(\mu_i; \complex)}
     \]
 
     so $U_i$ extends into an isometry between $\ol{A\xi_i}$ and $L^2(\mu_i; \complex)$. 
@@ -265,13 +265,13 @@
     
     Finally, given that
     \begin{enumerate}[label=(\roman*)]
-        \item By \hyperref[Goldstine's Theorem]{corollary:weak-dense-unit-ball}, $C(X; \complex)$ is weak*-dense in $[l^\infty(I); L^\infty(\mu_i; \complex)]$. 
+        \item By \hyperref[Goldstine's Theorem]{corollary:weak-dense-unit-ball}, $C(\Omega(A); \complex)$ is weak*-dense in $[l^\infty(I); L^\infty(\mu_i; \complex)]$. 
         \item The weak* topology on $[l^\infty(I); L^\infty(\mu_i; \complex)]$ is equal to the weak operator topology of $[l^\infty(I); L^\infty(\mu_i; \complex)]$ acting on $[l^2(I); L^2(\mu_i; \complex)]$. 
         \item $A$ is weak-operator dense in $B$. 
         \item By \hyperref[Spectral Theorem I]{theorem:spectral-theorem-vn-1}, the isomorphism $\phi \mapsto \int \phi dE$ is continuous from the weak* topology on $\mathscr{E}^* = [l^\infty(I); L^\infty(\mu_i; \complex)]$ to the weak operator topology on $B$.
     \end{enumerate}
     
-    the mapping $U$ is an unitary equivalence between $[l^\infty(I); L^\infty(\mu_i; \complex)]$ acting on $[l^2(I); L^2(\mu_i; \complex)]$ and $B$ acting on $H$. 
+    the mapping $U$ is a unitary equivalence between $[l^\infty(I); L^\infty(\mu_i; \complex)]$ acting on $[l^2(I); L^2(\mu_i; \complex)]$ and $B$ acting on $H$. 
 \end{proof}
 
 
@@ -292,10 +292,10 @@
         \item $I_E: \mathscr{E}^* \to A$ is the unique weak* to weak-operator continuous unital *-homomorphism such that $I_E(\text{Id}) = T$. 
     \end{enumerate}
     
-    The mapping $f \mapsto f(T)$ on $\mathscr{E}^*$ is the \textbf{$L^\infty$-functional calculus} of $T$. 
+    The spectral measure $E$ is the \textbf{resolution of the identity} for $T$, and the mapping $f \mapsto f(T)$ on $\mathscr{E}^*$ is the \textbf{$L^\infty$-functional calculus} of $T$. 
 \end{definition}
 \begin{proof}
-    (1), (2): By the \hyperref[Spectral Theorem]{theorem:spectral-theorem-vn-1} applied to $B(H)[T]$, there exists a unique spectral measure $E$ on $\sigma_{B(H)}(T)$ such that the mapping
+    (1), (2): By \hyperref[Spectral Theorem I]{theorem:spectral-theorem-vn-1} applied to $B(H)[T]$, there exists a unique spectral measure $E$ on $\sigma_{B(H)}(T)$ such that the mapping
     \[
     I_E: C(\sigma_{B(H)}(T); \complex)^{**} \to A \quad \phi \mapsto \int_{\sigma_{B(H)}(T)} \phi dE
     \]