I unfortunately deleted Charles's reply to the question about dimension being an operation. But thinking about it I realized that dimension is not an operation in the theory of vector spaces either; It is not preserved by morphisms. For vector spaces, even finite dimensional ones, the existence of dimension is a theorem. But the original question was not about vector spaces, but about coordinate spaces. For which the only morphisms are square permutation matrices. And the distributive law does not say that multiplication is a morphism with respect to addition. It does say that multiplication by a fixed element (on the right or the left) is a morphism with respect to addition. But I don't think that an infinitary distributive law (say between infinite sups and infinite infs) can be stated in such a way at all. Michael