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Homework Help: Span of a linearly independent subset of a hilbert space is a subspace iff finite

  1. Mar 5, 2012 #1
    1. The problem statement, all variables and given/known data

    Let S be a linearly independent subset of a Hilbert space. Prove that span(S) is a subspace, that is a linear manifold and a closed set, if and only if S is finite.

    2. Relevant equations

    3. The attempt at a solution

    Assuming S is finite means that S is a closed set (a finite subset of a metric space is closed). I think that this will help to prove span(S) is a closed set but I am a bit stuck.
  2. jcsd
  3. Mar 5, 2012 #2
    Sorry I have no idea
    Last edited: Mar 5, 2012
  4. Mar 5, 2012 #3


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    Science Advisor
    Homework Helper

    span(S) is the set of all FINITE linear combinations of elements in S. To see how that would make a problem for the set being closed if S is infinite, define a convergent series that contains multiples of an infinite number of elements of S.
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