- , with if and only if ,
- (the inner product is commutative),
- , and
- for all , .
One consequence of the inner product axioms is that the inner product is multilinear in both variables; that is:
An inner product space is a vector space together with an inner product.
- Anton, Howard and Chris Rorres. Elementary Linear Algebra: Applications Version. 9th ed. N.p.:John Wiley & Sons, Inc., 2005. p. 296