2
$\begingroup$

Need to know how to prove that every linear map is continuous. working with finite dimensional vector spaces. It's one of the problems I'm currently working on as revision for an upcoming test.

I know that that a map is continuous if the preimage of any open set is open but can't seem to figure out the full proof.

  • 1
    It's good that you can't figure out the proof, since the claim is not true.2012-11-13
  • 1
    Are you working with finite dimensional vector spaces?2012-11-13
  • 0
    @Sigur Yes i am. Chris, why do you claim that it's not true? Thanks2012-11-13
  • 0
    It's only false in infinite-dimensional vector spaces, Denis.2012-11-13
  • 0
    Thanks @KevinCarlson How would I go about solving it in finite dimensional vector spaces?2012-11-13

1 Answers 1

4

Here is the proof for finite dimensional vector spaces:

http://en.wikipedia.org/wiki/Discontinuous_linear_map#If_a_linear_map_is_finite_dimensional.2C_the_linear_map_is_continuous

  • 0
    Appreciated! Can't believe I didn't find this all along. Thanks2012-11-13