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Math Help - State on classical space

  1. #1
    Member Mauritzvdworm's Avatar
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    State on classical space

    I am looking for an example of a state (positive linear functional of norm 1) on the space C(\mathbb{T}^2) of continuous functions on the torus \mathbb{T}^2.

    My first guess would be something like

    \omega(f)=\int^{2\pi}_{0}\int^{2\pi}_{0}f(x,y)dxdy

    with norm

    \|\omega\|=\sup_{f\in C(\mathbb{T}^2)}\left|\int^{2\pi}_{0}\int^{2\pi}_{  0}f(x,y)dxdy\right|

    with f\in C(\mathbb{T}^2). However, if we take the function h to map any element of the torus identically to 1 the above definition would imply that

    \omega(h)=\int^{2\pi}_{0}\int^{2\pi}_{0}h(x,y)dxdy

    =\int^{2\pi}_{0}\int^{2\pi}_{0}dxdy

    =4\pi^2

    so this idea does not work. Any ideas?
    Last edited by Mauritzvdworm; June 23rd 2011 at 03:40 AM.
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  2. #2
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    Re: State on classical space

    What text are you using?
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  3. #3
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    Re: State on classical space

    Following your ideas, |w(f)| < 4(pi)^2 ||f|| where the last norm is the one in your space so what about v(f)=w(f)/4(pi)^2?
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  4. #4
    Member Mauritzvdworm's Avatar
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    Re: State on classical space

    I don't think that will work. It will be fine for that particular function, but we can just as well consider a function that maps any element of the torus to say 2, then the integral will have twice that value. I think I have an idea how to construct such a state, will post it if it works.
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  5. #5
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    Re: State on classical space

    I was assuming there was a typo in your definition of the norm of the functional since by linearity that quantity will never be bounded. So I assumed you meant \| \omega \| = \sup_{\| f \| =1} |\omega (f)| and for this definition my example does work.
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