# Advanced Linear Algebra Presentation

#### Stacey Lynn Baxter-Rienstra

**email**

###### sbaxter@cs.rice.edu ; rienstra@gauss.cl.uh.edu

Note: Due to the nature of HTML publishing the inner product notation will be square brackets instead of the pointed brackets.

**Lemma**

Let (V,|| ||) be a normed vector space.

Then || || comes from an inner product <=> the norm satisfies the *parallelogram law*

**|| u + v ||^2 + || u-v||^2 = 2(||u||^2 + ||v||^2) **

for all u,v in V.

## =>If (V,|| ||) is a normed vector space
by definition

V is a real vector space with a function

|| ||:V->R satisfying

- ||rv||= ([rv,rv])^(1/2) = ((r^2))^(1/2)([v,v])^(1/2) = |r| ||v||.
**positive definiteness: **||v||>= 0, and ||v|| = 0 <=> v = 0- ||u+v|| <= ||u|| + ||v||

## RHS

If || || comes from an inner product then:

(||u+v||)^2 +(||u-v||)^2

[u+v,u+v] + [u-v,u-v]
** using the definition of a norm**

[u,u]+[v,v]+[u,v]+[u,v]+[u,u]+[v,v]-[v,u]-[u,v]
** using bilinearity of [,] and symmetry.**

[u,u]+[v,v]+[u,u]+[v,v]

2[u,u]+2[v,v]

2([u,u]+[v,v])
**by using distributive property of R**

2(||u||^2 + ||v||^2)
**by definition of a norm of a vector space**

### Therefore, the Lemma has been proven in the forward direction.