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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.

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    Presumably you'd want it to have, at minimum, the property that the natural bi-linear map $h:A\times B \rightarrow A\otimes B$ is continuous. Additionally, you'd probably want, for any continuous morphism $A\times B\rightarrow V$, the induced map $A\otimes B\rightarrow V$ to be continuous.2012-01-20
  • 0
    Sorry, should be "for any continous bi-linear function $A\times B\rightarrow V$..."2012-01-20

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