fa.functional analysis - Is there a use for a Hilbert space that uses a different norm than the one induced by the inner product?

One place where this is used is with Hilbert bundles. Taking $L^2$-functions on a space generally works very badly, so one instead takes $L^{2,1}$-functions - functions which are differentiable (almost everywhere) with square-integrable first derivative. However, the transition functions aren't isometries with respect to this norm so one does tend to use the $L^2$-norm on this space, remembering that the fibres aren't complete with respect to the norm.

Another use is in Wiener integration. Depending on one's point of view, one either has a Hilbert space with a weaker norm, or a Banach space with a dense subspace equipped with a Hilbert norm. Essentially, having the two norms means that one can "tame" stuff with respect to one norm by using the stronger one.

This extends further to the notion of "rigged Hilbert spaces".

Generally, the idea is that you want to work in one space but you don't have enough control over convergence, so you introduce a stronger norm and then ordinary convergence with respect to the stronger norm implies fantastic convergence with respect to the weaker one.

(I apologise for using vague language, but the question doesn't give much indication of how deep an answer to give. For those who know a little about ideals of operators, one generally wants the inclusion from the strong norm to the weak to be at least trace class, and often Hilbert-Schmidt.)