Talk:Topological tensor product

Projective Norm = Largest Cross Norm?
In the section "Cross norms and tensor products of Banach spaces" it is stated: "There is a largest cross norm $$\pi$$ called the projective cross norm, [...]".

Why is the projective norm the largest cross norm?

Alternatively, one can clearly define a largest cross norm. In formula this may be given by $$\sup\{\nu(x):\nu\}$$ where the supremum runs over and exists due to all norms $$\nu$$ being cross. But here the equality of the projective norm and the largest cross norm is not obvious to me.

In case someone has an answer, would that person be so kind and add some comments or reference on this aspect? In case there doesn't find itself an answer, I suggest to remove "largest" as not justified.

Best reagrds, --Freeze S (talk) 15:20, 7 August 2018 (UTC)

Discussion of topology
It would seem fitting that this article should include a discussion of the projective topology that results when taking the tensor product? Also, I suppose it would be useful to mention that the symmetric tensor product is isomorphic(?) to the corresponding space of homogeneous polynomials. And I'm somewhat unclear on this, but the topology on the polynomials, which is given by the inverse limit of continuous seminorms, is in full agreement with the projective topology on the tensor product, right? linas (talk) 04:19, 8 June 2009 (UTC)

Modules instead of vector spaces
If we are dealing with modules over topological rings rather than topological vector spaces, is there a generalization of the topological tensor product? All monoidal categories have a tensor product defined for them, so what is the categorial tensor product when we are dealing with the category of topological spaces? --Moly 17:06, 6 October 2017 (UTC) — Preceding unsigned comment added by Moly (talk • contribs)