Definition:Inner Product/Complex Field

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

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

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

Also see

 * Definition:Complex Inner Product Space


 * Definition:Real Inner Product


 * Definition:Complex 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 complex inner products and real inner products