Completion Theorem (Inner Product Spaces)
Theorem
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.
Proof
the definition Definition:Completion (Inner Product Space) may be justified
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Sources
- John B. Conway: A Course in Functional Analysis (1990)... (previous)... (next) $I.1.9$
