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Homework Help: Banach space problem

  1. Nov 12, 2008 #1
    1. The problem statement, all variables and given/known data

    Let E be a Banach space and let M be a closed subspace of E. A
    vector x in E is called e-orthogonal to M if for all y in M the following inequality holds: ||x+y||>= (1- e)||x||.

    Prove that for each e>0 any proper subspace of M contains e-orthogonal

    3. The attempt at a solution

    Well clearly 0 is always such an element but how to find more elements?
    Last edited: Nov 12, 2008
  2. jcsd
  3. Nov 14, 2008 #2


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    What is e? A positive number less than 1? And did you really mean to say that M contains e-orthogonal elements? Because the only such possible element is 0. Proof: if x is in M and x is e-orthogonal to M, then ||x||(1-e) <= ||x+(-x)||=0.

    Edit: And if you meant to say "E contains elements which are e-orthogonal to M", then this result is known as (or at least immediately follows from) the Riesz Lemma.
    Last edited: Nov 14, 2008
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