Sorry to get back late; I've been out of town. Yes, I am regarding bicommutative Hopf algebras as abelian group objects in the category of cocommutative coalgebras, and they are internalizing the usual construction of tensor products of abelian groups.

Okay, since an answer which did not address the functional analysis has been accepted, that seals the impression that this is really a CW thread.

That reminds me of the story about Vopenka's principle: that Vopenka originally intended to demonstrate the absurdity of some large cardinal axioms, and introduced this "principle" with an eye to refuting it. But this backfired, and instead it's been studied enthusiastically. en.wikipedia.org/wiki/Vop%C4%9Bnka%27s_principle

The answers make this appear to be a CW thread, even if there is more functional analytic content in the question itself.

If memory serves, Kelley in his famous topology book does look at uniform function spaces.

@LSpice Thanks; fixed.

I’m voting to close this question because it was asked and answered at Math.SE: math.stackexchange.com/questions/4098045/…

Yay! Glad to see you're in the running!