140 lines
3.4 KiB
Markdown
140 lines
3.4 KiB
Markdown
# Lecture 25
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## Chapter VI Inner Product Spaces
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### Inner Products and Norms 6A
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#### Dot Product (Euclidean Inner Product)
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$$
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v\cdot w=v_1w_1+...+v_n w_n
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$$
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$$
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-\cdot -:\mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}
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$$
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Some properties
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* $v\cdot v\geq 0$
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* $v\cdot v=0\iff v=0$
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* $(u+v)\cdot w=u\cdot w+v\cdot w$
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* $(c\cdot v)\cdot w=c\cdot(v\cdot w)$
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#### Definition 6.2
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An inner product $\langle,\rangle:V\times V\to \mathbb{F}$
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Positivity: $\langle v,v\rangle\geq 0$
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Definiteness: $\langle v,v\rangle=0\iff v=0$
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Additivity: $\langle u+v,w\rangle=\langle u,w\rangle+\langle v,w\rangle$
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Homogeneity: $\langle \lambda u, v\rangle=\lambda\langle u,v\rangle$
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Conjugate symmetry: $\langle u,v\rangle=\overline{\langle v,u\rangle}$
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Note: the dot product on $\mathbb{R}^n$ satisfies these properties
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Example:
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$V=C^0([-1,-])$
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$L_2$ - inner product.
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$\langle f,g\rangle=\int^1_{-1} f\cdot g$
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$\langle f,f\rangle=\int ^1_{-1}f^2\geq 0$
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$\langle f+g,h\rangle=\langle f,h\rangle+\langle g,h\rangle$
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$\langle \lambda f,g\rangle=\lambda\langle f,g\rangle$
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$\langle f,g\rangle=\int^1_{-1} f\cdot g=\int^1_{-1} g\cdot f=\langle g,f\rangle$
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The result is in real vector space so no conjugate...
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#### Theorem 6.6
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For $\langle,\rangle$ an inner product
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(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).
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(b,c) $\langle 0,v\rangle=\langle v,0\rangle=0$
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(d) $\langle u,v+w\rangle=\langle u,v\rangle+\langle u,w\rangle$ (second terms are additive.)
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(e) $\langle u,\lambda v\rangle=\bar{\lambda}\langle u,v\rangle$
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#### Definition 6.4
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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.
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For the remainder of the chapter. $V,W$ are inner product vector spaces...
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#### Definition 6.7
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For $v\in V$ the **norm of $V$** is given by $||v||:=\sqrt{\langle v,v\rangle}$
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#### Theorem 6.9
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Suppose $v\in V$.
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(a) $||v||=0\iff v=0$
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(b) $||\lambda v||=|\lambda|\ ||v||$
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Proof:
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$||\lambda v||^2=\langle \lambda v,\lambda v\rangle =\lambda\langle v,\lambda v\rangle=\lambda\bar{\lambda}\langle v,v\rangle$
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So $|\lambda|^2 \langle v,v\rangle=|\lambda|^2||v||^2$, $||\lambda v||=|\lambda|\ ||v||$
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#### Definition 6.10
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$v,u\in V$ are **orthogonal** if $\langle v,u\rangle=0$.
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#### Theorem 6.12 (Pythagorean Theorem)
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If $u,v\in V$ are orthogonal, then $||u+v||^2=||u||^2+||v||$
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Proof:
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$$
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\begin{aligned}
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||u+v||^2&=\langle u+v,u+v\rangle\\
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&=\langle u,u+v\rangle+\langle v,u+v\rangle\\
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&=\langle u,u\rangle+\langle u,v\rangle+\langle v,u\rangle+\langle v,v\rangle\\
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&=||u||^2+||v||^2
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\end{aligned}
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$$
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#### Theorem 6.13
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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.
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#### Theorem 6.14 (Cauchy-Schwarz)
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Let $u,v\in V$, then $|<u,v>|\leq ||u||\ ||v||$ where equality occurs only $u,v$ are parallel...
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Proof:
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Take the square norm of $u=\frac{<u,v>}{||u||^2}v+w$.
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#### Theorem 6.17 Triangle Inequality
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If $u,v\in V$, then $||u+v||\leq ||u||+||v||$
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Proof:
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$$
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\begin{aligned}
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||u+v||^2&=<u+v,u+v>\\
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&=<u,u>+<u,v>+<v,u>+<v,v>\\
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&=||u||^2+||v||^2+2Re(<u,v>)\\
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&\leq ||u||^2+||v||^2+2|<u,v>|\\
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&\leq ||u||^2+||v||^2+2||u||\ ||v||\\
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&\leq (||u||+||v ||)^2
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\end{aligned}
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$$
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