Fixed typos in quadratic.

src/fa/norm/hilbert.tex

@@ -373,7 +373,7 @@ A significant property of Hilbert spaces is that every closed subspace is comple
 \label{definition:hilbert-quadratic-function}
     Let $H$ be a complex Hilbert space and $q: H \to \complex$, then $q$ is \textbf{quadratic} if:
     \begin{enumerate}
-        \item For each $\lambda \in \complex$ and $\xi \in H$, $q(\lambda x) = |\lambda|^2 q(x)$. 
+        \item For each $\lambda \in \complex$ and $\xi \in H$, $q(\lambda x) = |\lambda|^2 q(\xi)$. 
         \item (\textbf{Parallelogram Law}) For each $\xi, \eta \in H$, 
         \[
         q(\xi + \eta) + q(\xi - \eta) = 2q(\xi) + 2q(\eta)
@@ -420,15 +420,15 @@ A significant property of Hilbert spaces is that every closed subspace is comple
 
         \item For each $\xi, \eta, \theta \in H$, 
         \[
-        \dpn{\xi + \theta, \eta}{H} = \dpn{\xi, \eta}{H} + \dpn{\theta, \eta}{H}
+        \dpn{\xi + \theta, \eta}{q} = \dpn{\xi, \eta}{q} + \dpn{\theta, \eta}{q}
         \]
 
         and
         \[
-        \dpn{\xi, \eta + \theta}{H} = \dpn{\xi, \eta}{H} + \dpn{\xi, \theta}{H}
+        \dpn{\xi, \eta + \theta}{q} = \dpn{\xi, \eta}{q} + \dpn{\xi, \theta}{q}
         \]
 
-        \item For each dyadic rational number $\lambda \in \mathbb{D}$ with $\lambda \in \mathbb{D}$, 
+        \item For each dyadic rational number $\lambda \in \mathbb{D}$, 
         \[
         \dpn{\lambda\xi, \eta}{q} = \lambda\dpn{\xi, \eta}{q} = \dpn{\xi, \lambda \eta}{q}
         \]
@@ -446,7 +446,7 @@ A significant property of Hilbert spaces is that every closed subspace is comple
         4\dpn{\xi, \eta}{q} + 4 \dpn{\theta, \eta}{q}  = \sum_{k = 0}^3 i^k\braks{q(\xi + i^k\eta) + q(\theta + i^k\eta)}
     \]
 
-    For each $0 \le k \le 3$, by the Parallelogram Law applied to the pair $((\xi + \theta)/2 + i^k\eta, (\xi - \eta)/2)$, 
+    For each $0 \le k \le 3$, by the Parallelogram Law applied to the pair $((\xi + \theta)/2 + i^k\eta, (\xi - \theta)/2)$, 
     \[
     q(\xi + i^k\eta) + q(\theta + i^k\eta) = 2\braks{q\paren{\frac{\xi + \theta}{2} + i^k\eta} + q\paren{\frac{\xi - \theta}{2}}}
     \]
@@ -455,7 +455,7 @@ A significant property of Hilbert spaces is that every closed subspace is comple
     \begin{align*}
         4\dpn{\xi, \eta}{q} + 4 \dpn{\theta, \eta}{q}  &= \sum_{k = 0}^3 i^k\braks{q(\xi + i^k\eta) + q(\theta + i^k\eta)} \\ 
         &= 2\sum_{k = 0}^3 i^k\braks{q\paren{\frac{\xi + \theta}{2} + i^k\eta} + q\paren{\frac{\xi - \theta}{2}}} \\
-        &= 2\sum_{k = 0}^3 i^k{q\paren{\frac{\xi + \theta}{2} + i^k\eta}} = 2 \angles{\frac{\xi + \theta}{2}, \eta}_{q}
+        &= 2\sum_{k = 0}^3 i^k{q\paren{\frac{\xi + \theta}{2} + i^k\eta}} = 8 \angles{\frac{\xi + \theta}{2}, \eta}_{q}
     \end{align*}
 
     Similarly, for each $0 \le k \le 3$, by the Parallelogram Law applied to the pair $(\xi + i^k(\eta + \theta)/2, i^k(\eta - \theta)/2)$,
@@ -469,7 +469,7 @@ A significant property of Hilbert spaces is that every closed subspace is comple
     \]
     
 
-    so $\dpn{\cdot, \cdot}{q}: H \times H \to \complex$ is a $\real$-bilinear form. 
+    
 
     (1.ii): By definition of $\dpn{\cdot, \cdot}{q}$, the claim holds for $0$ and $1$. Let $n \in \natz$ and suppose inductively that the claim holds for all $\lambda \in \mathbb{D}_n \cap [0, 1]$. Take $\lambda \in \mathbb{D}_{n+1} \cap [0, 1]$, then there exists $\mu, \nu \in \mathbb{D}_n \cap [0, 1]$ such that $\lambda = (\mu + \nu)/2$. In which case, by (1.i),
     \begin{align*}
@@ -495,8 +495,8 @@ A significant property of Hilbert spaces is that every closed subspace is comple
     
     (1.v): Let $\xi, \eta \in H$, then
     \begin{align*}
-        |\dpn{\xi, \eta}{q}| &\le \frac{1}{4}\sum_{k = 0}^3 |q(\xi + i^k\eta)| = \frac{M}{4}\sum_{k = 0}^3\normn{\xi + i^k\eta}_H \\
+        |\dpn{\xi, \eta}{q}| &\le \frac{1}{4}\sum_{k = 0}^3 |q(\xi + i^k\eta)| \le \frac{M}{4}\sum_{k = 0}^3\normn{\xi + i^k\eta}_H^2 \\
-        &\le \frac{M}{4}\sum_{k = 0}^3 [\norm{\xi}_H^2 + \norm{\eta}_H^2 + i^{-k}\dpn{\xi, \eta}{H}] = M(\norm{\xi}_H^2 + \norm{\eta}_H^2)
+        &\le \frac{M}{4}\sum_{k = 0}^3 [\norm{\xi}_H^2 + \norm{\eta}_H^2 + 2\text{Re}\dpn{i^{-k}\xi, \eta}{H}] = M(\norm{\xi}_H^2 + \norm{\eta}_H^2)
     \end{align*}
     
     
@@ -510,7 +510,7 @@ A significant property of Hilbert spaces is that every closed subspace is comple
     (1.vii): By (1.iv) and (1.vi), $\dpn{\cdot, \cdot}{q}$ is a $\real$-bilinear form on $H$. For any $\xi, \eta \in H$, by absolute homogeneity of $q$, 
     \begin{align*}
         \dpn{i\xi, \eta}{q} &= \frac{1}{4}\sum_{k = 0}^3 i^k q(i\xi + i^k\eta) = \frac{1}{4}\sum_{k = 0}^3 i^k q(\xi + i^{k-1}\eta) \\
-        &= \frac{i}{4}\sum_{k = 0}^3 i^{k-1} q(\xi + i^{k-1}\eta) = \dpn{\xi, \eta}{q} \\
+        &= \frac{i}{4}\sum_{k = 0}^3 i^{k-1} q(\xi + i^{k-1}\eta) = i\dpn{\xi, \eta}{q} \\
         \dpn{\xi, i\eta}{q} &= \frac{1}{4}\sum_{k = 0}^3 i^k q(\xi + i^{k+1}\eta) = \frac{1}{4}\sum_{k = 0}^3 i^k q(i^{-1}\xi + i^{k}\eta) \\
         &= \dpn{-i\xi, \eta}{q} =  -i \dpn{\xi, \eta}{q}
     \end{align*}