NoteNextra-origin/pages/Math429/Math429_L25.md

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}