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Interpreting a problem on Frechet spaces (topology)

  1. Dec 11, 2011 #1
    1. The problem statement, all variables and given/known data
    I've been given the following problem:

    "Suppose that U is a finite-dimensional subspace of a Fréchet space [itex](V,\tau)[/itex]. Show that the subspace topology on U is the usual topology (given for example by a Euclidean norm) and that U is a closed linear subspace of V."

    I feel a bit confused: I'm not sure what precisely is meant by 'the usual topology (given for example by a Euclidean norm)', but IIRC any 2 norms on a finite dimensional vector space are equivalent (to e.g. the Euclidean norm) so am I right in thinking that the question just requires me to show that the subspace topology on U is a topology induced by some norm (and therefore equivalent to any norm on U)?

    I suppose to show closure is fairly simple, just play around with bases and limits, and it's a subspace of a vector space so linearity is also trivial, right? So then it's only the first part which I need help with, if I understand everything else correctly - thanks!
     
  2. jcsd
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