Are most matrices invertible? discusses this question for matrices. All the answers implicitly used the (unique) vector topology on the space of $n \times n$ matrices. But my understanding (correct me if I'm wrong) is that infinite-dimensional linear operators can have multiple vector norms which induce inequivalent topologies, so the question becomes trickier. My (not entirely precise) question is, for an infinite-dimensional vector space, is the set of isomorphisms a dense open set under every "reasonable" operator topology? Under every operator norm topology induced by a "reasonable" norm on the vector space? My intuition says yes, but I'd be curious if anyone can make the words "reasonable" more precise.
(I assume that there's no natural way to generalize the Lebesgue-measure sense of "almost all matrices are invertible" to the infinite-dimensional case, due to the absence of an inifinite-dimensional Lebesgue measure, but correct me if I'm wrong.)