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at general linear group we only had some sentences on its incarnation as an algebraic group. I have started a subsection Definition – As a topological group with some basics.
What’s a quick proof that the topology on as a subspace of with its Euclidean topology coincides with that as a subspace of with its compact-open topology?
I guess ?
So we do have an injective continuous map , the currying of the continuous action . This says that the usual Euclidean topology is finer than the subspace topology coming from .
But the Euclidean topology is also coarser. Let’s do this for instead of . A Euclidean neighborhood base of a linear map or matrix consists of sets of the form . But this is a basis/base element for the function space topology, where and is the -ball about .
(It may help to think of convergence in the function space topology as the exact same as uniform convergence over every compact set.)
Thanks, Todd!
I have added that to the entry here.
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