I heard several times (for instance in these general lectures) that Grothendieck did *functional analysis* before he started doing algebraic geometry and category theory. It is said that at the time he did functional analysis, he didn't yet know what a category is, but his thinking was already very category-theoretic. For instance, I think McLarty roughly said in the linked lecture (although I don't recall the time stamp) that Grothendieck implicitly came up himself with the notion of a universal property and characterized some notions of tensor product in functional analysis via universal properties.

**Question:** Which theorems concretely did Grothendieck prove in functional analysis which are "category-theoretic" in spirit? How do these statements connect to more traditional functional analysis? (What I mean by the second question is for example: when he proved that some tensor products have a specific universal property, in which way did he apply that to solve concrete problems in functional analysis?)

I know that this question is not very concrete and rather historical (I apologize for that -- added the soft-question tag), but it's something I was wondering about when listening to these lectures, and maybe someone knows something interesting and more concrete to say.