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Inner Product Space/Hilbert Space Problem

  1. Jun 17, 2010 #1
    1. The problem statement, all variables and given/known data
    3. If z is any fixed element of an inner product space X, show that f(x) = <x,z> defines a bounded linear functional f on X, of norm ||z||.
    4. Consider Prob. 3. If the mapping X --> X' (the space of continuous linear functionals) given by z |--> f is surjective, show that X must be a Hilbert space.

    2. Relevant equations



    3. The attempt at a solution
    I solved question 3 without any difficulty, but I can't seem to make any progress on question 4.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jun 17, 2010 #2
    To start you off, write down explicitly what it means for a map to be surjective and write down the requirements for something to be a Hilbert space.

    What are your ideas about showing that the map z |--> f is surjective?

    How would you show each of the requirements for a Hilbert space?

    Coto
     
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