# 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:

$$
\begin{aligned}
    ||u+v||^2&=<u+v,u+v>\\
    &=<u,u>+<u,v>+<v,u>+<v,v>\\
    &=||u||^2+||v||^2+2Re(<u,v>)\\
    &\leq ||u||^2+||v||^2+2|<u,v>|\\
    &\leq  ||u||^2+||v||^2+2||u||\ ||v||\\
    &\leq (||u||+||v ||)^2
\end{aligned}
$$
