Typo fixes.

src/op/system/complete-upgrade.tex

@@ -94,7 +94,7 @@ The following is a slight generalisation of one form of the above example, but s
     \norm{\phi}_{\text{cb}} &\le \norm{\phi}_{S^*}
     \end{align*}
 
-    (2): If $\xi = \eta$, then
+    (2): If $\xi = \eta$, then for any $a \in A$ with $a \ge 0$, 
     \[
     \sum_{i, j = 1}^n a_{ij}\eta_j\ol{\eta_i} = (\xi \otimes \one)^* \cdot a \cdot (\xi \otimes \one) \ge 0
     \]
@@ -221,7 +221,7 @@ The following is a slight generalisation of one form of the above example, but s
 
 \begin{example}[Reduction Map]
 \label{example:matric-reduction-map}
-    Let $\phi: M_n(\complex) \to M_n(\complex)$ be defined by
+    Let $n \ge 2$ and $\phi: M_n(\complex) \to M_n(\complex)$ be defined by
     \[
     \phi(x) = \text{Tr}(x)1_{M_n(\complex)} - x
     \]

src/op/system/dilation.tex

@@ -308,7 +308,7 @@
 
     so $(K, U)$ is a unitary power dilation of $T$. 
     
-    (U): By the \hyperref[Stone-Weierstrass Theorem]{theorem:complex-stone-weierstrass}, $(\pi, V, K)$ is minimal if and only if $\bigcup_{n \in \integer}U^n H$ is dense in $K$, if and only if $U$ admits no non-trivial reducing subspace. 
+    (U): By the \hyperref[Stone-Weierstrass Theorem]{theorem:complex-stone-weierstrass}, $(\pi, V, K)$ is minimal if and only if the span of $\bigcup_{n \in \integer}U^n H$ is dense in $K$, if and only if $U$ admits no non-trivial reducing subspace. 
     
     By \hyperref[uniqueness of minimal Stinespring representations]{proposition:stinespring-minimal-uniqueness}, such a dilation is unique. 
 \end{proof}
@@ -444,7 +444,7 @@
     \normn{p(T(z_1, \cdots, z_n))}_{M_n(\complex)} = \dpn{p(T(z_1, \cdots, z_n))\xi, \eta}{\complex^n} = f_{\xi, \eta}(z_1, \cdots, z_n)
     \]
 
-    By maximality, $f_{\xi, \eta}$ achieves its maximum modulus at $(z_1, \cdots, z_n)$, so (1) implies that $(z_1, \cdots, z_n) \in \partial B$. At which point, $|z_j| = 1$ for all $1 \le j \le n$, and $\text{diag}(z_1, \cdots, z_n)$ is unitary. Therefore $T(z_1, \cdots, z_n) = U\text{diag}(z_1, \cdots, z_n)V$ is also unitary. 
+    By maximality, $f_{\xi, \eta}$ achieves its maximum modulus at $(z_1, \cdots, z_n)$, so (1) implies that $|z_j| = 1$ for all $1 \le j \le n$ and $\text{diag}(z_1, \cdots, z_n)$ is unitary. Therefore $T(z_1, \cdots, z_n) = U\text{diag}(z_1, \cdots, z_n)V$ is also unitary. 
 
     (3): Since $(s_1, \cdots, s_n) \in B$, (2) implies that  
     \begin{align*}

src/op/system/schur.tex

@@ -116,7 +116,7 @@
     \begin{align*}
         A &= (I - R)^{-1} + (I - R^*)^{-1} - I \\
         &= (I - R)^{-1}(I - R^* + I - R - (I - R)(I - R)^*)(I - R^*)^{-1} \\
-        &= (I - R)^{-1}(I - RR^*)(1 - R^*)^{-1} \ge 0
+        &= (I - R)^{-1}(I - RR^*)(I - R^*)^{-1} \ge 0
     \end{align*}
     
 \end{proof}