# Connexions

You are here: Home » Content » Digital Signal Processing » Properties of Inner Products

### Recently Viewed

This feature requires Javascript to be enabled.

Inside Collection (Course):

Course by: Mark A. Davenport. E-mail the author

# Properties of Inner Products

Module by: Mark A. Davenport. E-mail the author

Inner products and their induced norms have some very useful properties:

• Cauchy-Schwartz Inequality: |x,y|xy|x,y|xy with equality iff αCαC such that y=αxy=αx
• Pythagorean Theorem: x , y = 0 x + y 2 = x - y 2 = x 2 + y 2 x , y = 0 x + y 2 = x - y 2 = x 2 + y 2
• Parallelogram Law: x + y 2 + x - y 2 = 2 x 2 + 2 y 2 x + y 2 + x - y 2 = 2 x 2 + 2 y 2
• Polarization Identity: Re [ x , y ] = x + y 2 - x - y 2 4 Re [ x , y ] = x + y 2 - x - y 2 4

In R2R2 and R3R3, we are very familiar with the geometric notion of an angle between two vectors. For example, if x,yR2x,yR2, then from the law of cosines, x,y=xycosθx,y=xycosθ. This relationship depends only on norms and inner products, so it can easily be extended to any inner product space.

## Definition 1

The angleθθ between two vectors x,yx,y in an inner product space is defined by cosθ=x,yxycosθ=x,yxy

## Definition 2

Vectors x,yx,y in an inner product space are said to be orthogonal if x,y=0x,y=0.

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

#### Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

#### Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks