TITLE: Topology on the tensor product of two topological vector spaces -- how properties does it maintains?
QUESTION [9 upvotes]: Good morning, this is my first question in this website. If I have two topological vector spaces, say $A$ and $B$, I would like to know
1)how the topology on $A\otimes B$ is canonically defined?
2)if the topologies on $A$ and $B$ are locally convex, so is the topology on $A\otimes B$?
I'm doing this question because if I have a compact Lie group $G$ and two $G$-modules $A$ and $B$, I want the "averaging operator" to be defined on $A$, $B$ and also $A\otimes B$. I found that the topology should be locally convex and 'feebly complete' (how "The structure of compact groups" (Hofmann-Morris) calls it), so I would like to know if these properties hold in the tensor topology.

REPLY [8 votes]: AFAIK there is no single "canonical" topology on $A \otimes B$. Instead there are several that make sense.
A good source is

Francois Treves: "Topological Vector Spaces, Distributions and Kernels"

chapter 43: "The Two Main Topologies on Tensor Products", where the author defines the $\epsilon$ and the $\pi$ topologies.
The $\pi$-topology is the strongest locally convext topology such that the canonical embedding of $A \times B \to A \otimes B$ is continuous (locally convex by definition).
( The definition of the $\epsilon$ topology is rather involved, so you better look it up in the book I mentioned.)