An inner product is a generalization of the dot product. In a vector space, it is a way to multiply vectors together, with the result of this multiplication being a scalar.
More precisely, for a real vector space, an inner product satisfies the following four properties. Let , , and be vectors and be a scalar, then:
1. .
2. .
3. .
4. and equal if and only if .
example:
For Euclidean space , the inner product is given by
satisfies the following four properties. Let 
, 
, and 
be vectors and 
be a scalar, then:
.
.
.
and equal if and only if 
.
, the inner product is given by 