Added states.

src/fa/norm/hilbert.tex

@@ -83,20 +83,23 @@
 
 \begin{definition}[Inner Product]
 \label{definition:inner-product}
-    Let $E$ be a vector space over $K$ and $\inp_E: E \times E \to K$, then $\inp_E$ is an \textbf{inner product} if:
+    Let $E$ be a vector space over $K$ and $\inp_E: E \times E \to K$, then $\inp_E$ is a \textbf{pseudo inner product} if:
     \begin{enumerate}[label=(H\arabic*)]
         \item For each $x, y, z \in E$, $\angles{x + y, z}_E = \dpn{x, z}{E} + \dpn{y, z}{E}$. 
         \item For any $x, y \in E$ and $\mu \in K$, $\dpn{\mu x, y}{E} = \mu \dpn{x, y}{E}$. 
         \item For every $x, y \in E$, $\dpn{x, y}{E} = \ol{\dpn{y, x}{E}}$. 
-        \item[(I)] For each $x \in E$, $\dpn{x, x}{E} \ge 0$, with equality if and only if $x = 0$. 
+        \item[(I)] For each $x \in E$, $\dpn{x, x}{E} \ge 0$. 
     \end{enumerate}
 
+    and an \textbf{inner product} if for each $x \in E$, $\dpn{x, x}{E} = 0$ if and only if $x = 0$. 
+    
+
 \end{definition}
 
 
 \begin{proposition}[Cauchy-Schwarz Inequality]
 \label{proposition:cauchy-schwarz}
-    Let $H$ be a vector space over $K \in \RC$ and $\inp_H: E \times E \to K$ be an inner product, then for any $x, y \in H$, $\dpn{x, y}{H} \le \norm{x}_H\norm{y}_H$. 
+    Let $H$ be a vector space over $K \in \RC$ and $\inp_H: E \times E \to K$ be a pseudo inner product, then for any $x, y \in H$, $\dpn{x, y}{H} \le \norm{x}_H\norm{y}_H$. 
 \end{proposition}
 \begin{proof}[Proof, {{\cite[Theorem 5.19]{Folland}}}. ]
     Assume without loss of generality that $\dpn{x, y}{H} > 0$, then for each $t \in \real$,