Fix typo.

src/fa/lc/hahn-banach.tex

@@ -26,10 +26,10 @@
 
     then for any $x \in F$ and $t > 0$,
     \begin{align*}
-        \phi(x + tx_0) &= \phi(x) + t\lambda = t\braks{\phi(t^{-1}x) + \lambda} \\
+        \phi_{x_0, \lambda}(x + tx_0) &= \phi(x) + t\lambda = t\braks{\phi(t^{-1}x) + \lambda} \\
         &\le t\braks{\rho(t^{-1}x + x_0) - \phi(t^{-1}x) + \phi(t^{-1}x)} \\
         &= t\rho(t^{-1}x + x_0) = \rho(x + tx_0) \\
-        \phi(x - tx_0) &= \phi(x) - t\lambda = t\braks{\phi(t^{-1}x) - \lambda} \\
+        \phi_{x_0, \lambda}(x - tx_0) &= \phi(x) - t\lambda = t\braks{\phi(t^{-1}x) - \lambda} \\
         &\ge t\braks{\rho(t^{-1}x - x_0) + \phi(t^{-1}x) - \phi(t^{-1}x)} \\
         &= t\rho(t^{-1}x - x_0) = \rho(x - tx_0)
     \end{align*}

src/op/banach/definitions.tex

@@ -29,9 +29,16 @@
 
 \begin{definition}[Unital Homomorphism]
 \label{definition:banach-algebra-unital-homomorphism}
-    Let $A, B$ be unital Banach algebras and $\phi: A \to B$ be a homomorphism, then $\phi$ is a \textbf{unital homomorphism} if $\phi(1) = 1$. 
+    Let $A, B$ be unital Banach algebras and $\phi: A \to B$ be a homomorphism, then $\phi$ is a \textbf{unital homomorphism} if $\phi(1_A) = 1_B$. 
 \end{definition}
 
+\begin{definition}[Representation]
+\label{definition:banach-algebra-representation}
+    Let $A$ be a Banach algebra, then a \textbf{representation} of $A$ is a pair $(E, \pi)$ where $E$ is a Banach space, and $\pi: A \to L(E; E)$ is a continuous homomorphism. 
+\end{definition}
+
+
+
 \begin{definition}[Unitisation]
 \label{definition:unitisation}
     Let $A$ be a Banach algebra over $\complex$, and $\tilde A = \complex \oplus A$ with

src/op/c-star/homomorphism.tex

@@ -54,4 +54,24 @@
     The above setup implies that for every $y \in \ol{\Phi(A)} \cap B_{sa}$, there exists $z \in A_{sa}$ such that $\norm{y - \Phi(z)}_{B} \le \norm{y}_B/2$, and $\norm{z}_A \le 2\norm{y}_B$. By the \hyperref[method of successive approximations]{theorem:successive-approximation}, $\phi(A_{sa}) = \ol{\Phi(A)} \cap B_{sa}$. Therefore $\Phi(A) = \ol{\Phi(A)}$. 
 \end{proof}
 
+\begin{definition}[Representation of $C^*$-Algebra]
+\label{definition:representation-cstar-algebra}
+    Let $A$ be a $C^*$-algebra, then a \textbf{representation} of $A$ is a pair $(H, \pi)$, where $H$ is a Hilbert space, and $\pi: A \to B(H)$ is a *-homomorphism. 
+\end{definition}
+
+\begin{definition}[Unitary Equivalence]
+\label{definition:representation-unitary-equivalent}
+    Let $A$ be a $C^*$-algebra and $(H_1, \pi_1), (H_2, \pi_2)$ be representations of $A$, then $(H_1, \pi_1)$ and $(H_2, \pi_2)$ are \textbf{unitarily equivalent} if there exists an isometry $U \in L(H_1; H_2)$ such that the following diagram commutes
+    \[
+    \xymatrix{
+    H_1 \ar@{->}[r]^{U} \ar@{->}[d]_{\pi_1(x)} & H_2 \ar@{->}[d]^{\pi_2(x)} \\
+    H_1 & H_2 \ar@{->}[l]^{U^*}
+    }
+    \]
+
+    for all $x \in A$. 
+\end{definition}
+
+
+