format updates

content/Math429/Math429_L25.md

@@ -23,17 +23,17 @@ Some properties
 
 #### Definition 6.2
 
-An inner product $<,>:V\times V\to \mathbb{F}$
+An inner product $\langle,\rangle:V\times V\to \mathbb{F}$
 
-Positivity: $<v,v>\geq 0$
+Positivity: $\langle v,v\rangle\geq 0$
 
-Definiteness: $<v,v>=0\iff v=0$
+Definiteness: $\langle v,v\rangle=0\iff v=0$
 
-Additivity: $<u+v,w>=<u,w>+<v,w>$
+Additivity: $\langle u+v,w\rangle=\langle u,w\rangle+\langle v,w\rangle$
 
-Homogeneity: $<\lambda u, v>=\lambda<u,v>$
+Homogeneity: $\langle \lambda u, v\rangle=\lambda\langle u,v\rangle$
 
-Conjugate symmetry: $<u,v>=\overline{<v,u>}$
+Conjugate symmetry: $\langle u,v\rangle=\overline{\langle v,u\rangle}$
 
 Note: the dot product on $\mathbb{R}^n$ satisfies these properties
 
@@ -43,39 +43,39 @@ $V=C^0([-1,-])$
 
 $L_2$ - inner product.
 
-$<f,g>=\int^1_{-1} f\cdot g$
+$\langle f,g\rangle=\int^1_{-1} f\cdot g$
 
-$<f,f>=\int ^1_{-1}f^2\geq 0$
+$\langle f,f\rangle=\int ^1_{-1}f^2\geq 0$
 
-$<f+g,h>=<f,h>+<g,h>$
+$\langle f+g,h\rangle=\langle f,h\rangle+\langle g,h\rangle$
 
-$<\lambda f,g>=\lambda<f,g>$
+$\langle \lambda f,g\rangle=\lambda\langle f,g\rangle$
 
-$<f,g>=\int^1_{-1} f\cdot g=\int^1_{-1} g\cdot f=<g,f>$
+$\langle f,g\rangle=\int^1_{-1} f\cdot g=\int^1_{-1} g\cdot f=\langle g,f\rangle$
 
 The result is in real vector space so no conjugate...
 
 #### Theorem 6.6
 
-For $<,>$ an inner product
+For $\langle,\rangle$ an inner product
 
-(a) Fix $V$, then the map given by $u\mapsto <u,v>$ is a linear map (Warning: if $\mathbb{F}=\mathbb{C}$, then $u\mapsto<u,v>$ is not linear).
+(a) Fix $V$, then the map given by $u\mapsto \langle u,v\rangle$ is a linear map (Warning: if $\mathbb{F}=\mathbb{C}$, then $u\mapsto\langle u,v\rangle$ is not linear).
 
-(b,c) $<0,v>=<v,0>=0$
+(b,c) $\langle 0,v\rangle=\langle v,0\rangle=0$
 
-(d) $<u,v+w>=<u,v>+<u,w>$ (second terms are additive.)
+(d) $\langle u,v+w\rangle=\langle u,v\rangle+\langle u,w\rangle$ (second terms are additive.)
 
-(e) $<u,\lambda v>=\bar{\lambda}<u,v>$
+(e) $\langle u,\lambda v\rangle=\bar{\lambda}\langle u,v\rangle$
 
 #### Definition 6.4
 
-An **inner product space** is a pair of vector space and inner product on it. $(v,<,>)$. In practice, we will say "$V$ is an inner product space" and treat $V$ as the vector space.
+An **inner product space** is a pair of vector space and inner product on it. $(v,\langle,\rangle)$. In practice, we will say "$V$ is an inner product space" and treat $V$ as the vector space.
 
 For the remainder of the chapter. $V,W$ are inner product vector spaces...
 
 #### Definition 6.7
 
-For $v\in V$ the **norm of $V$** is given by $||v||:=\sqrt{<v,v>}$
+For $v\in V$ the **norm of $V$** is given by $||v||:=\sqrt{\langle v,v\rangle}$
 
 #### Theorem 6.9
 
@@ -86,13 +86,13 @@ Suppose $v\in V$.
 
 Proof:
 
-$||\lambda v||^2=<\lambda v,\lambda v> =\lambda<v,\lambda v>=\lambda\bar{\lambda}<v,v>$
+$||\lambda v||^2=\langle \lambda v,\lambda v\rangle =\lambda\langle v,\lambda v\rangle=\lambda\bar{\lambda}\langle v,v\rangle$
 
-So $|\lambda|^2 <v,v>=|\lambda|^2||v||^2$, $||\lambda v||=|\lambda|\ ||v||$
+So $|\lambda|^2 \langle v,v\rangle=|\lambda|^2||v||^2$, $||\lambda v||=|\lambda|\ ||v||$
 
 #### Definition 6.10
 
-$v,u\in V$ are **orthogonal** if $<v,u>=0$.
+$v,u\in V$ are **orthogonal** if $\langle v,u\rangle=0$.
 
 #### Theorem 6.12 (Pythagorean Theorem)
 
@@ -102,9 +102,9 @@ Proof:
 
 $$
 \begin{aligned}
-    ||u+v||^2&=<u+v,u+v>\\
-    &=<u,u+v>+<v,u+v>\\
-    &=<u,u>+<u,v>+<v,u>+<v,v>\\
+    ||u+v||^2&=\langle u+v,u+v\rangle\\
+    &=\langle u,u+v\rangle+\langle v,u+v\rangle\\
+    &=\langle u,u\rangle+\langle u,v\rangle+\langle v,u\rangle+\langle v,v\rangle\\
     &=||u||^2+||v||^2
 \end{aligned}
 $$