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Math Help - Proving completeness in $L_2(E)$...

  1. #1
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    Proving completeness in $L_2(E)$...

    The journal I am reading through contains the following theorem...

    Theorem: The set $\{w_n(y,\tau)\}$ is linearly independent and complete in $L_2(E).$

    (If you are interested the work is on the heat equation and $w_n$ represents the fundamental solution given at a denumerable number of points and $E$ is the lateral surface of a cylinder under consideration).

    I am fine with the proof of linear independence, my question is how do you prove a sequence is complete in functional analysis in general?

    In the proof he lets an $f(y,\tau)$ be an arbitrary function in $E$ and
    $$\int_{0}^{T}d\tau\int_{S}f(y,\tau)w_n(y,\tau) \,dS = 0, \, n=1,2,3,...$$

    He then states that if $f(y,\tau) = 0$ then the sequence $\{w_n(y,\tau)\}$ is complete.

    Sorry if this is a stupid question above, just like details of why we do the above in the proof.
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  2. #2
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    Quote Originally Posted by KroneckerDelta View Post
    The journal I am reading through contains the following theorem...

    Theorem: The set $\{w_n(y,\tau)\}$ is linearly independent and complete in $L_2(E).$

    (If you are interested the work is on the heat equation and $w_n$ represents the fundamental solution given at a denumerable number of points and $E$ is the lateral surface of a cylinder under consideration).

    I am fine with the proof of linear independence, my question is how do you prove a sequence is complete in functional analysis in general?

    In the proof he lets an $f(y,\tau)$ be an arbitrary function in $E$ and
    $$\int_{0}^{T}d\tau\int_{S}f(y,\tau)w_n(y,\tau) \,dS = 0, \, n=1,2,3,...$$ Shouldn't that dS be dy?

    He then states that if $f(y,\tau) = 0$ then the sequence $\{w_n(y,\tau)\}$ is complete.

    Sorry if this is a stupid question above, just like details of why we do the above in the proof.
    The space L_2(E) is a Hilbert space. In a Hilbert space, the closed linear subspace spanned by a set W is equal to the second orthogonal complement W^{\perp\perp} of W. Clearly if W^\perp = \{0\} then W^{\perp\perp} will be the whole space. So to prove that a set W is complete, you need to show that W^\perp = \{0\}.

    To do that, you take an arbitrary element f\in L_2(E) and show that if the inner product \langle f,w\rangle=0 for all w in W then f must be 0.

    The equation \int_{0}^{T}d\tau\int_{S}f(y,\tau)w_n(y,\tau) \,dy = 0 is exactly the condition that that inner product should be 0.
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  3. #3
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    Thanks for the reply Opalg, really helpful and makes total sense now, I should most definately brush up on my linear algebra .
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