Oopsies

src/op/banach/definitions.tex

@@ -12,7 +12,7 @@
 
 \begin{definition}[Unital Banach Algebra]
 \label{definition:unital-banach-algebra}
-    Let $(A, \norm{\cdot}_A)$ be a Banach algebra, then $A$ is \textbf{unital} if there exists $1 \in A$ such that for any $x \in A$, $x1 = 1x = x$. In which case, there exists an equivalent norm $\norm{\cdot}_{1}: A \to [0, \inftu)$ such that $\norm{1}_1 = 1$, and $A$ is always assumed to be equipped with this norm.  
+    Let $(A, \norm{\cdot}_A)$ be a Banach algebra, then $A$ is \textbf{unital} if there exists $1 \in A$ such that for any $x \in A$, $x1 = 1x = x$. In which case, there exists an equivalent norm $\norm{\cdot}_{1}: A \to [0, \infty)$ such that $\norm{1}_1 = 1$, and $A$ is always assumed to be equipped with this norm.  
 \end{definition}
 \begin{proof}[Proof, {{\cite[Proposition I.1.3]{Takesaki1}}}. ]
     For each $x \in A$, let $L_x \in L(A; A)$ be defined by $y \mapsto xy$, and let $\norm{x}_1 = \norm{L_x}_{L(A; A)}$, then $\norm{x}_1 \le \norm{x}_A$ and $\norm{1}_1 = 1$. On the other hand, $\frac{\norm{x}_A}{\norm{1}_A} \le \norm{x}_1$, so $\norm{\cdot}_1$ is equivalent to $\norm{\cdot}_A$.
@@ -82,7 +82,7 @@
     The algebra $\tilde A$ is the \textbf{unitisation} of $A$. 
 \end{definition}
 \begin{proof}
-    (U): For each $\lambda + x \in \tilde A$, let $\tilde \phi(\lambda + x) = \lamdba + \phi(x)$. 
+    (U): For each $\lambda + x \in \tilde A$, let $\tilde \phi(\lambda + x) = \lambda + \phi(x)$. 
 \end{proof}