# Completion Theorem (Inner Product Space)

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## 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.

## Proof

This theorem requires a proof.In particular: the definition Definition:Completion (Inner Product Space) may be justifiedYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Sources

- 1990: John B. Conway:
*A Course in Functional Analysis*(2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 1.$ Elementary Properties and Examples: Proposition $1.9$