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Caracterizing a subspace of L^2

  1. Apr 5, 2008 #1

    quasar987

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    [SOLVED] Caracterizing a subspace of L^2

    1. The problem statement, all variables and given/known data

    Call M the subspace of L²([0,1]) consisting of all functions of vanishing mean. I.e., [itex]u\in M \Rightarrow \int_0^1u(s)ds=0[/itex].

    I am trying to find the dimension of the orthogonal of M,

    [tex]M^{\perp}=\{x\in L^2([0,1]):\int_0^1x(s)u(s)ds=0 \ \forall u\in M\}[/tex]

    I would be surprised if [itex]M^{\perp}[/itex] was anything other than the constant functions, but my attempts at a proof have been unsuccessful.

    Any idea how to go at this?
     
  2. jcsd
  3. Apr 8, 2008 #2

    quasar987

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    Solved.
     
    Last edited: Apr 8, 2008
  4. Apr 8, 2008 #3

    morphism

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    Was it the (a.e.) constant functions in the end?

    Sorry I couldn't help -- been busy with studying for my finals!
     
  5. Apr 8, 2008 #4

    Dick

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    How could it be anything else? The set of vanishing mean functions is basically defined as the set of all u(x) such that <u(x),1>=0. L^2 is a norm up to a.e.
     
  6. Apr 8, 2008 #5

    morphism

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    Well, yeah... :tongue2:
     
  7. Apr 8, 2008 #6

    Dick

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    Nice to talk to an erudite and sophisticated gentleman for a change. Instead of a bonehead. :)
     
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