Real Inner Product is Complex Inner Product

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

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

 * Real Inner Product Space is Complex Inner Product Space


 * Real Semi-Inner Product is Complex Semi-Inner Product