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content/Math429/Math429_L25.md

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+# Lecture 25
+
+## Chapter VI Inner Product Spaces
+
+### Inner Products and Norms 6A
+
+#### Dot Product (Euclidean Inner Product)
+
+$$
+v\cdot w=v_1w_1+...+v_n w_n
+$$
+
+$$
+-\cdot -:\mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}
+$$
+
+Some properties
+
+* $v\cdot v\geq 0$  
+* $v\cdot v=0\iff v=0$  
+* $(u+v)\cdot w=u\cdot w+v\cdot w$
+* $(c\cdot v)\cdot w=c\cdot(v\cdot w)$
+
+#### Definition 6.2
+
+An inner product $<,>:V\times V\to \mathbb{F}$
+
+Positivity: $<v,v>\geq 0$
+
+Definiteness: $<v,v>=0\iff v=0$
+
+Additivity: $<u+v,w>=<u,w>+<v,w>$
+
+Homogeneity: $<\lambda u, v>=\lambda<u,v>$
+
+Conjugate symmetry: $<u,v>=\overline{<v,u>}$
+
+Note: the dot product on $\mathbb{R}^n$ satisfies these properties
+
+Example:
+
+$V=C^0([-1,-])$
+
+$L_2$ - inner product.
+
+$<f,g>=\int^1_{-1} f\cdot g$
+
+$<f,f>=\int ^1_{-1}f^2\geq 0$
+
+$<f+g,h>=<f,h>+<g,h>$
+
+$<\lambda f,g>=\lambda<f,g>$
+
+$<f,g>=\int^1_{-1} f\cdot g=\int^1_{-1} g\cdot f=<g,f>$
+
+The result is in real vector space so no conjugate...
+
+#### Theorem 6.6
+
+For $<,>$ 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).
+
+(b,c) $<0,v>=<v,0>=0$
+
+(d) $<u,v+w>=<u,v>+<u,w>$ (second terms are additive.)
+
+(e) $<u,\lambda v>=\bar{\lambda}<u,v>$
+
+#### 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.
+
+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 $||v||:=\sqrt{<v,v>}$
+
+#### Theorem 6.9
+
+Suppose $v\in V$.
+
+(a) $||v||=0\iff v=0$  
+(b) $||\lambda v||=|\lambda|\ ||v||$
+
+Proof:
+
+$||\lambda v||^2=<\lambda v,\lambda v> =\lambda<v,\lambda v>=\lambda\bar{\lambda}<v,v>$
+
+So $|\lambda|^2 <v,v>=|\lambda|^2||v||^2$, $||\lambda v||=|\lambda|\ ||v||$
+
+#### Definition 6.10
+
+$v,u\in V$ are **orthogonal** if $<v,u>=0$.
+
+#### Theorem 6.12 (Pythagorean Theorem)
+
+If $u,v\in V$ are orthogonal, then $||u+v||^2=||u||^2+||v||$
+
+Proof: 
+
+$$
+\begin{aligned}
+    ||u+v||^2&=<u+v,u+v>\\
+    &=<u,u+v>+<v,u+v>\\
+    &=<u,u>+<u,v>+<v,u>+<v,v>\\
+    &=||u||^2+||v||^2
+\end{aligned}
+$$
+
+#### Theorem 6.13
+
+Suppose $u,v\in V$, $v\neq 0$, set $c=\frac{<u,v>}{||v||^2}$, then let $w=u-v\cdot v$, then $v$ and $w$ are orthogonal.
+
+#### Theorem 6.14 (Cauchy-Schwarz)
+
+Let $u,v\in V$, then $|<u,v>|\leq ||u||\ ||v||$ where equality occurs only $u,v$ are parallel...
+
+Proof:
+
+Take the square norm of $u=\frac{<u,v>}{||u||^2}v+w$.
+
+#### Theorem 6.17 Triangle Inequality
+
+If $u,v\in V$, then $||u+v||\leq ||u||+||v||$
+
+Proof:
+
+$$
+\begin{aligned}
+    ||u+v||^2&=<u+v,u+v>\\
+    &=<u,u>+<u,v>+<v,u>+<v,v>\\
+    &=||u||^2+||v||^2+2Re(<u,v>)\\
+    &\leq ||u||^2+||v||^2+2|<u,v>|\\
+    &\leq  ||u||^2+||v||^2+2||u||\ ||v||\\
+    &\leq (||u||+||v ||)^2
+\end{aligned}
+$$