When I first introduced the notions of algebraically complete categories (every endo-functor has an initial algebra) and algebraically compact categories (every endo-functor has an initial algebra naturally isomorphic to a terminal co-algebra) I thought that such things would exist in the classical foundations only in degenerate form. It was a surprise when I noticed that the categories of countable sets and of countably dimensioned vector spaces are algebraically complete. I'm now surprised by: The category of separable Hilbert spaces and linear operators of bound at most 1 is algebraically compact. As in the earlier cases one doesn't really need the controlling cardinal number to be aleph-naught. To avoid using the axiom of choice one can state the more general result by taking an arbitrary Hilbert space A and defining *A* by: Objects of *A*: all those Hilbert spaces that can be isometrically embedded in A; Maps of *A*: all linear operators thereon of bound at most 1. Then: *A* is algebraically compact. The theorem holds for both the real and complex cases. In the proof I use the surprising (to me) fact that in the categories in question every half-invertible map has a _unique_ half-inverse. (That is, every map has at most one left-inverse and one right- inverse.) All the other examples from nature that I can think of are categories in which the only half-invertibles are invertible. Are there others? Peter Freyd