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.

Proof

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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 $1.9$