logic - Is the (first order theory) of Hilbert spaces categorical?

Suppose the axioms of a Hilbert space (i.e. vector space, inner product, completeness and separability) are formulated as a first order theory.

It can be shown that any infinite dimensional Hilbert space is isomorphic to $l^2$ (the space of infinite sequences of complex numbers the sum of whose absolute squares converges). In other words, any model satisfying the (infinite dimensional) Hilbert space axioms is isomorphic to $l^2$.

1. Since all the models are isomorphic to $l^2$ (and hence to each other), the Hilbert space theory is categorical.




2. But by Lowenheim-Skolem theorem, any first order theory that has an infinite model of cardinality c, also has a model of any cardinality larger c.

So 2 seems to contradict 1, since all models of a categorical theory have to have the same cardinality (in order to be isomorphic).

How is this resolved?