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Math Help - a question on completeness within functional analysis

  1. #1
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    Question a question on completeness within functional analysis

    suppose I have a function f(x,t) where both x and t are continuous, I treat x as variable, but t as parameter, as t varies, I get a sequence of f(x), I want to know whether this sequence of f(x) is complete to build up all possible functions on x.

    This should be a well-defined problem, but maybe the tones are unclear. First, should I introduce say L2 norm when talking about completeness? Second, can anybody lead me to look into this problem?

    Thanks,
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  2. #2
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    What do you mean by "completeness" as applied to a sequence? In functional analysis, a space may be complete (all Cauchy sequences converge) but that term is not normally applied to individual sequences.
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    What I mean completeness is for the functional space defined on 1-dimensional real number. What I want to know is how to determine whether the sequence of functions actually form a complete set for that functional space defined on the 1d real number. The criterion for all Cauchy sequences to converge in the same space seems to be hard to use for this purpose. For example, the set of functions by e^(ikx) is complete for the 1D real numbers, but this does not seem to me to be related with convergence of Cauchy sequences.
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  4. #4
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    Quote Originally Posted by bsmile View Post
    What I mean completeness is for the functional space defined on 1-dimensional real number. What I want to know is how to determine whether the sequence of functions actually form a complete set for that functional space defined on the 1d real number. The criterion for all Cauchy sequences to converge in the same space seems to be hard to use for this purpose. For example, the set of functions by e^(ikx) is complete for the 1D real numbers, but this does not seem to me to be related with convergence of Cauchy sequences.
    So you mean like a complete set in the sense that the closure of the linear span is the whole space (like a complete orthonormal set in a Hilbert space)? If so then it depends heavily on which sets (both the space and your subset of functions) you're talking about. Do you have a concrete example of what you're asking?
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  5. #5
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    for example,

    K(x;\omega,T)=\frac{\pi}{\omega} [2\omega \coth(\frac{x}{2T}) - (\omega + x) \coth(\frac{\omega + x}{2T}) + (\omega - x) \coth(\frac{\omega - x}{2T})]

    the metric space the functional is defined is the normed space, viz,  L^{2}(\Omega). x ranges over the whole 1D space, \omega=(2 n +1)\pi T where n is all integers, T is all positive real numbers.
    Last edited by bsmile; January 4th 2010 at 03:17 PM.
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  6. #6
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    Quote Originally Posted by bsmile View Post
    for example,

    K(x;\omega,T)=\frac{\pi}{\omega} [2\omega \coth(\frac{x}{2T}) - (\omega + x) \coth(\frac{\omega + x}{2T}) + (\omega - x) \coth(\frac{\omega - x}{2T})]

    where x is the 1D space, \omega and T are continuous parameters with constraint T>0.
    Okay so we know the family of functions but in what space do they live in ( H^1(\Omega ), \ C^{\infty } (\Omega ), \ C(\Omega ), \ L^2 (\Omega ), \ L^p (\Omega )?) , over what domain \Omega \subset \mathbb{R} are this functions defined. These are all important questions that you need to have an answer for before you even begin to think about a problem like this.
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    Thanks for giving me more hints on the problem!

    the metric space in which the functions are defined should be the usual normed space, viz.  L^{2}(\Omega), and the domain for x is the whole real space, \omega=(2 n +1)\pi T where n is all integers, T is all positive real numbers.
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  8. #8
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    Quote Originally Posted by Jose27 View Post
    Okay so we know the family of functions but in what space do they live in ( H^1(\Omega ), \ C^{\infty } (\Omega ), \ C(\Omega ), \ L^2 (\Omega ), \ L^p (\Omega )?) , over what domain \Omega \subset \mathbb{R} are this functions defined. These are all important questions that you need to have an answer for before you even begin to think about a problem like this.
    Thanks for giving me more hints on the problem!

    the metric space in which the functions are defined should be the usual normed space, viz.  L^{2}(\Omega), and the domain for x is the whole real space, \omega=(2 n +1)\pi T where n is all integers, T is all positive real numbers.
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  9. #9
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    Quote Originally Posted by bsmile View Post
    Thanks for giving me more hints on the problem!

    the metric space in which the functions are defined should be the usual normed space, viz.  L^{2}(\Omega), and the domain for x is the whole real space, \omega=(2 n +1)\pi T where n is all integers, T is all positive real numbers.
    Okay, this doesn't look easy. First, if I undertood what you were asking we need to prove that \overline{ span (A) } = L^2 (\mathbb{R} ) (?) where A is your set of functions. So we need to show that for any given function f \in L^2 ( =L^2(\mathbb{R} ) ) and every \epsilon >0 there exists a g \in span(A) such that \Vert f-g \Vert < \epsilon. Right now I don't see how you could prove this assertion, let me think it for a bit and get back to you.
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  10. #10
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    Quote Originally Posted by Jose27 View Post

    Quote Originally Posted by bsmile View Post
    K(x;\omega,T)=\frac{\pi}{\omega} [2\omega \coth(\frac{x}{2T}) - (\omega + x) \coth(\frac{\omega + x}{2T}) + (\omega - x) \coth(\frac{\omega - x}{2T})]

    the metric space in which the functions are defined should be the usual normed space, viz.  L^{2}(\Omega), and the domain for x is the whole real space, \omega=(2 n +1)\pi T where n is all integers, T is all positive real numbers.
    Okay, this doesn't look easy. First, if I undertood what you were asking we need to prove that \overline{ span (A) } = L^2 (\mathbb{R} ) (?) where A is your set of functions. So we need to show that for any given function f \in L^2 ( =L^2(\mathbb{R} ) ) and every \epsilon >0 there exists a g \in span(A) such that \Vert f-g \Vert < \epsilon. Right now I don't see how you could prove this assertion, let me think it for a bit and get back to you.
    Sorry, I don't understand the meaning of \overline{ span (A) } , but what I mean is for any function f(x) which belongs to the vector space spanned by K(x;w,T) and satisfies \int_{-\infty}^{\infty}|f(x)|^2 \mathrm{d}x<\infty, what you assumed should hold.

    Maybe the difficulty lies in that \omega is discrete? Then, we can try to show first for \omega \ne 0 whether the completeness condition holds.

    One observation is that as x \to \infty, K(x;\omega,T) \to 0, this might show that for finite \omega and T,span(K) cannot be complete for all functions in the 1D real space(they cannot form functions with finite value at x=infinity). But it is still unclear if \omega and T can be infinite.
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