Inner product space

An inner product on a vector space is a function , and satisfies the following rules:

•  
•  and 
• 

For all  and . 

It is important to note that any inner product on the vector space creates a norm on the said vector space, which we see as follows:

We can notice from these rules and definitions that all inner product spaces are also normed spaces, and therefore also metric spaces.

Another very important concept is orthogonality, which in a nutshell means that two vectors are perpendicular to each other (that is, they are at a right angle to each other) from Euclidean space. 

Two vectors are orthogonal if their inner product is zero—that is, . As a shorthand for perpendicularity, we write . 

Additionally, if the two orthogonal vectors are of unit length—that is, , then they are called orthonormal.

In general, the inner product in  is as follows: