Completion Theorem (Inner Product Space)

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

Let $\left\langle{\cdot, \cdot}\right\rangle_V$ 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 $\left\langle{\cdot, \cdot}\right\rangle_V$ 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.