Hilbert space

A real Hilbert space is a real inner-product space $H$ whose induced-length metric is complete: every Cauchy sequence in $H$ for the norm $\mathrm{\parallel }v\mathrm{\parallel }=\sqrt{⟨v,v⟩}$ converges to a point of $H$. A complex Hilbert space is a complex inner-product space with the same completeness property, so that a Hilbert space is exactly a real or complex inner-product space that is a Banach space for its induced norm (Banach space).

The completion convention is the Cauchy-sequence one. Banach space defines completeness by convergence of Cauchy sequences, and this page keeps that convention throughout. It is weaker than $\sigma$-completeness — the assertion that every decreasing sequence of nonempty closed subsets with diameters tending to zero has a nonempty intersection. The two agree in ZFC, but Blackadar, Farah and Karagila note that the closest-point theorem on a $\sigma$-complete inner-product space is provable in ZF, whereas the Cauchy-complete form used below consumes the Axiom of Countable Choice; nothing here silently imports the stronger notion.

Real and complex inner-product spaces and their induced length

Complete metric space: every Cauchy sequence converges in the space

The induced length is a norm

Banach space