Real Semi-Inner Product is Complex Semi-Inner Product

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Theorem

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

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

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

Proof

This follows immediately from:

• Definition of Real Semi-Inner Product

• Definition of Complex Semi-Inner Product

• Complex Number equals Conjugate iff Wholly Real

$\blacksquare$

Also see

• Real Semi-Inner Product Space is Complex Semi-Inner Product Space

• Real Inner Product is Complex Inner Product