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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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    Sorry, should be "for any continous bi-linear function $A\times B\rightarrow V$..."2012-01-20

1 Answers 1

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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.)