Definition:Inner Product/Real Field

Definition
Let $V$ be a vector space over a real subfield $\GF$.

A (real) inner product is a mapping $\innerprod \cdot \cdot: V \times V \to \GF$ that satisfies the real inner product axioms:

That is, a (real) inner product is a real semi-inner product with the additional condition $(4)$.

Also see

 * Definition:Real Inner Product Space


 * Definition:Complex Inner Product


 * Definition:Real Semi-Inner Product, a slightly more general concept.


 * The most well-known example of an inner product is the dot product (see Dot Product is Inner Product).


 * Definition:Hilbert Space


 * Inner Product/Examples for examples of both real inner products and complex inner products