Housekeeping.

src/vna/vn/index.tex

@@ -8,5 +8,6 @@
 \input{./commutative.tex}
 \input{./spec.tex}
 \input{./fc.tex}
+\input{./unlimited.tex}
 \input{./projection.tex}
-\input{./type-decomp.tex}
+\input{./type-decomp.tex}

src/vna/vn/spec.tex

@@ -24,10 +24,14 @@
     \begin{enumerate}
         \item For each $x, y \in H$, $\norm{E_{x, y}}_{\text{var}} \le \norm{x}_H \norm{y}_H$. 
         \item For each $x \in H$, $E_{x, x}$ is positive.
+        \item For each $x, y \in H$ and $B \in \cb_X$, 
+        \[
+        |E_{x, y}(B)| \le E_{x, x}(B)^{1/2} E_{y, y}(B)^{1/2} \le \frac{1}{2}(E_{x, x}(B) + E_{y, y}(B))
+        \]
     \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=2]
+    \begin{enumerate}[start=3]
         \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 
         \[
@@ -50,7 +54,17 @@
 
     (2): For each $B \in \cb_X$, $E(B)$ is a projection, so $E_{x, x}(B) = \dpn{E(B)x, x}{H} \ge 0$.
 
-    (3): For each $x, y \in H$ and $B, C \in \cb_X$, 
+    (3): For each $B \in \cb_X$, by the \hyperref[Cauchy-Schwarz inequality]{proposition:cauchy-schwarz},
+    \begin{align*}
+        |E_{x, y}(B)| &= |\dpn{E(B)x, y}{H}| = |\dpn{E(B)x, E(B)y}{H}| \\
+        &\le \dpn{E(B)x, E(B)x}{H}^{1/2}\dpn{E(B)y, E(B)y}{H}^{1/2} \\
+        &= E_{x, x}(B)^{1/2} E_{y, y}(B)^{1/2}
+    \end{align*}
+
+    By \hyperref[Young's Inequality]{lemma:young-inequality}, $E_{x, x}(B)^{1/2} E_{y, y}(B)^{1/2} \le \frac{1}{2}(E_{x, x}(B) + E_{y, y}(B))$. 
+    
+
+    (4): For each $x, y \in H$ and $B, C \in \cb_X$, 
     \[
     \int_C \one_B dE_{x, y} = \dpn{E(C \cap B)x, y}{H} = \dpn{E(C)E(B)x, y}{H} = E_{E(B)x, y}(C)
     \]
@@ -59,7 +73,7 @@
     
     Finally, let $\mu \in \mathscr{E}$ and $\nu \in M_R(X; \complex)$ with $\nu \ll \mu$, then by the \hyperref[Radon-Nikodym Theorem]{theorem:lebesgue-radon-nikodym}, there exists $f \in L^1(\mu; \complex)$ such that $d\nu = f d\mu \in \mathscr{E}$. 
 
-    (4): By (3), for any $\mu \in \mathscr{E}$ and $f \in L^1(\mu; \complex)$, $fd\mu \in \mathscr{E}$ as well. By \autoref{proposition:measures-dual-algebra}, there exists a unique weak*-continuous involution and separately weak*-continuous product on $\mathscr{E}^*$ making $\mathscr{E}^*$ a commutative unital $C^*$-algebra, and $J|_{C(X; \complex)}$ a unital *-homomorphism. Since
+    (5): By (4), for any $\mu \in \mathscr{E}$ and $f \in L^1(\mu; \complex)$, $fd\mu \in \mathscr{E}$ as well. By \autoref{proposition:measures-dual-algebra}, there exists a unique weak*-continuous involution and separately weak*-continuous product on $\mathscr{E}^*$ making $\mathscr{E}^*$ a commutative unital $C^*$-algebra, and $J|_{C(X; \complex)}$ a unital *-homomorphism. Since
     \begin{enumerate}[label=(\roman*)]
         \item $J$ is $\sigma(B^\infty(X; \complex), M_R(X; \complex))$-$\sigma(\mathscr{E}^*, \mathscr{E})$ continuous. 
         \item Conjugation on $B^\infty(X; \complex)$ is $\sigma(B^\infty(X; \complex), M_R(X; \complex))$-continuous.
@@ -300,7 +314,7 @@
 \begin{proof}
     Let $E$ be the spectral measure on $\Omega(A)$ associated with $A$, and $\mathscr{E} \subset M_R(\Omega(A); \complex)$ be the closed subspace generated by $\bracsn{E_{x, y}|x, y \in H}$. By \hyperref[Spectral Theorem I]{theorem:spectral-theorem-vn-1}, $A$ is *-isomorphic to $\mathscr{E}^*$.
 
-    (1): By (3) of \autoref{lemma:spectral-measure-properties} and \autoref{theorem:hilbert-measures-dual}, $A$ is *-isomorphic to $[l^\infty(I); L^\infty(\mu_i; \complex)]$, where $\seqi{\mu} \subset \mathscr{E}$ is a maximal mutually singular family. Let $\Omega = \bigsqcup_{i \in I}\Omega(A)$, then $\Omega$ is a LCH space. For each $i \in I$, let $\Omega_i$ denote the $i$-th copy of $\Omega(A)$, then
+    (1): By (4) of \autoref{lemma:spectral-measure-properties} and \autoref{theorem:hilbert-measures-dual}, $A$ is *-isomorphic to $[l^\infty(I); L^\infty(\mu_i; \complex)]$, where $\seqi{\mu} \subset \mathscr{E}$ is a maximal mutually singular family. Let $\Omega = \bigsqcup_{i \in I}\Omega(A)$, then $\Omega$ is a LCH space. For each $i \in I$, let $\Omega_i$ denote the $i$-th copy of $\Omega(A)$, then
     \[
     \mu: \cb_\Omega \to [0, \infty] \quad B \mapsto \sum_{i \in I}\mu_i(B \cap \Omega_i)
     \]