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We show that no faithful functor from the category of Hilbert spaces withlinear isometries into the category of sets preserves directed colimits. ThusHilbert spaces cannot form an abstract elementary class, even up to change oflanguage. We deduce an analogous result for the category of commutative unital$C^\ast$-algebras with $\ast$-homomorphisms. This implies, in particular, thatthis category is not axiomatizable by a first-order theory, a strengthening ofa conjecture of Bankston.

Hilbert spaces and ${C}^\ast$-algebras are not finitely concrete