Real Inner Product is Complex Inner Product

Theorem
Let $\R$ be the field of real numbers.

Let $\GF$ be a subfield of $\R$.

Let $V$ be a vector space over $\GF$.

Let $\innerprod \cdot \cdot: V \times V \to \GF$ be a real inner product.

Then $\innerprod \cdot \cdot$ is a complex inner product

Proof
This follows immediately from:






 * Complex Number equals Conjugate iff Wholly Real

Also see

 * User:Leigh.Samphier/Refactor/Real Inner Product Space is Complex Inner Product Space


 * User:Leigh.Samphier/Refactor/Real Semi-Inner Product is Complex Semi-Inner Product