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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.

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I am by no means an expert in the field of functional analysis, but I found some references that might be interesting and could serve as an answer (perhaps better called a soft answer). The following article by F. Bombal describes some of Grothendieck's work on functional analysis.

One of the main theorems mentioned is the Kernel Theorem. Grothendieck's approach to this theorem was via nuclear spaces, creating a general framework from which the theorem follows as something self-evident. As I understand it, this is where Grothendieck used the universal properties of the tensor products the OP mentions. About Grothendieck's proof of the Kernel Theorem, Bombal writes:

By the way, this is a typical example of Grothendieck's way of doing mathematics: put the problem in a more general setting and find a general theory (usually, very deep and far-reaching) which contains the solution of the initial problem as a particular case.

In another passage, Bombal makes more explicit mention of functorial methods in Grothendieck's Sur Les Applications Lineaires Faiblement Compactes D'Espaces Du Type C(K):

... the article emphasizes the "functorial" point of view of Grothendieck: In order to study the structure of some mathematical object, you have to look at the behavior of the morphisms on and into it. This was quite usual in some parts of Mathematics, but not so in Analysis.

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