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Prove: sum of a finite dim. subspace with a subspace is closed

  1. Jun 4, 2012 #1
    1. The problem statement, all variables and given/known data
    Prove:
    If ##X## is a (possibly infinite dimensional) locally convex space, ##L \leq X##, ##dimL < \infty ##, and ##M \leq X ## then ##L + M## is closed.



    2. Relevant equations



    3. The attempt at a solution

    ##dimL < \infty \implies L## is closed in ##X##
    ##L+M = \{ x+y : x\in L, y \in M \} \implies ^{??} dim(L+M) < \infty \implies L+M ## is closed in ##X##
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jun 4, 2012 #2

    micromass

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    Don't you need M to be closed as well??

    Anyway, your attempt isn't correct since L+M doesn't need to be finite-dimensional.
     
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