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.