  We study the functor l^2 from the category of partial injections to the
category of Hilbert spaces. The former category is finitely accessible, and its
homsets are algebraic domains; the latter category has conditionally algebraic
domains for homsets. The functor preserves daggers, monoidal structures,
enrichment, and various (co)limits, but has no adjoints. Up to unitaries, its
direct image consists precisely of the partial isometries, but its essential
image consists of all continuous linear maps between Hilbert spaces.
