Completion Theorem (Inner Product Space)

Theorem

Let $V$ be an inner product space over a subfield $\GF$ of $\C$.

Let $\innerprod \cdot \cdot_V$ be the inner product on $V$.

Let $d: V \times V \to \R_{\ge 0}$ be the metric induced by the inner product norm.

Let $H$ be the completion of $V$ with respect to $d$.

Then $\innerprod \cdot \cdot_V$ can be extended to an inner product on $H$.

By definition, $H$ will be a Hilbert space.

Therefore, the theorem can alternatively be stated as:

Any inner product space may be completed to a Hilbert space.