TITLE: Has Abstract Algebra ever been of service to Analysis?
QUESTION [20 upvotes]: I’m not saying that it ought to be. I was just wondering whether it has. What I have in mind is that it would have been of material help in proving, say, the Hahn-Banach Theorem, or some such. If it has, what is the most important/impressive instance of this?

REPLY [5 votes]: Developing analysis beyond Euclidean space (e.g., on manifolds that don't a priori have a natural embedding into $\mathbb R^n$) requires a fair dose of multilinear algebra in order to define differential forms, which are the objects that are integrated. I think integration on manifolds counts as something of basic interest to analysts. :)