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Timothy Murphy
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 Posted: Sat Jul 05, 2008 8:01 am Post subject: Compact-open topology |
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Given locally compact (and Hausdorff) spaces X,Y
let C(X,Y) be the space of continuous maps X->Y
with the compact-open topology.
Is there some simple condition on X,Y
that will ensure that C(X,Y) is locally compact? |
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Timothy Murphy
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| Posted: Tue Jul 08, 2008 11:14 am Post subject: Re: Compact-open topology |
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Mariano Suárez-Alvarez wrote:
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Of course we know that the dual group G* of a locally compact abelian
group is locally compact.
I was wondering if this was a particular case of a larger class
of function spaces known to be locally compact.
Eg is the space of homomorphisms from one locally compact abelian group
to another known to be locally compact?
But the dual group G* is *not* the complete
mapping space, but a very small part of it.
If you are willing to consider *subspaces*
of mapping spaces, there are lots and lots of
locally compact (and even compact, of course)
ones. Even non trivial ones.
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I realise of course that G* is not C(G,T).
Just to explain the context.
I'm giving a course on group representations next academic year,
and I thought I would consider locally compact abelian groups.
It seems quite difficult to show that if A is such a group
then A* (with the compact-open topology) is locally compact,
and the proofs I have seen do not seem to go far beyond A*.
This made me wonder if there was a wider setting for the result.
Eg is C(A,T) locally compact?
Or is the group of homomorphisms A->B locally compact,
if B is locally compact?
Or perhaps if B is compact?
I haven't really thought about it seriously,
and was just looking for pointers. |
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