I've been studying some category theory lately and in particular, I became acquainted with the notions of products and coproducts, which led me to ponder the following:
Consider the category of all complex Hilbert spaces (the morphisms being linear isometries). This category has coproducts, due to the direct sum construction: if $X_{\alpha} , {\alpha\in\Lambda}$ is family of Hilbert spaces, define $X := \bigoplus_{\alpha\in\Lambda}X_{\alpha}$ as the set of all "$\Lambda$-tuples" $(x_\alpha)_{\alpha \in \Lambda}$ such that:
$x_\alpha \in X_\alpha \\ \forall \alpha$ and $\sum_{\alpha \in \Lambda} \|x_{\alpha}\|^2 < \infty$
Then one can define addition, scalar multiplication and an inner product on $X$ in an obvious way, and we have the canonical inclusion maps.
However, I don't see any way to make this construction into a product, though maybe there is another construction I don't know of.
I'm sorry if this question is elementary for category theorists, but to me it's not so obvious.
EDIT: Thanks for the replies. As it was pointed out, this category doesn't even admit finite products with morphisms being linear isometries. As I don't see any more natural choice for morphisms, I suppose there isn't any good answer to my question (other than "no" :)).
Best Answer
The category you specified does not have products, because it doesn't have a product of zero objects. The product of zero objects is an final object, if it exists, but final objects need to have unique maps (in this case, isometries) from all other objects. Such a final Hilbert space would need to be at least as large as any other Hilbert space (hence nonzero), but it cannot have nonidentity automorphisms (such as minus one).