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Thread: Completeness of Vector Space

  1. #1
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    Completeness of Vector Space

    Let $\displaystyle H$ be the space of sequences $\displaystyle (x(1),x(2),x(3),...)$ such that $\displaystyle \sum^{\infty}_{j=1}\frac{|x(j)|^2}{j^2}<\infty$, with inner product $\displaystyle <x,y>=\sum^{\infty}_{j=1}\frac{x(j)\overline{y(j)} }{j^2}$ and norm $\displaystyle \|x\|=<x,x>^{1/2}$.

    I need to show that $\displaystyle H$ is complete with respect to the inner product i.e. a Hilbert Space. I am assuming that $\displaystyle \mathbb{C}$ is complete.

    I know that I need to show that every Cauchy sequence in H is convergent. I can see that this follows relatively easily from the definition of H.

    So I have taken a sequence $\displaystyle x_{n}(j)$ but for this to be Cauchy $\displaystyle |x_{n}(j)-x_{m}(j)|<\epsilon$. But I can't see how to show that, given that the inner product on $\displaystyle H$ has a $\displaystyle j^2$ below it. Any help would be great. Thanks
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  2. #2
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    Quote Originally Posted by ejgmath View Post
    Let $\displaystyle H$ be the space of sequences $\displaystyle (x(1),x(2),x(3),...)$ such that $\displaystyle \sum^{\infty}_{j=1}\frac{|x(j)|^2}{j^2}<\infty$, with inner product $\displaystyle <x,y>=\sum^{\infty}_{j=1}\frac{x(j)\overline{y(j)} }{j^2}$ and norm $\displaystyle \|x\|=<x,x>^{1/2}$.

    I need to show that $\displaystyle H$ is complete with respect to the inner product i.e. a Hilbert Space. I am assuming that $\displaystyle \mathbb{C}$ is complete.

    I know that I need to show that every Cauchy sequence in H is convergent. I can see that this follows relatively easily from the definition of H.

    So I have taken a sequence $\displaystyle x_{n}(j)$ but for this to be Cauchy $\displaystyle |x_{n}(j)-x_{m}(j)|<\epsilon$. But I can't see how to show that, given that the inner product on $\displaystyle H$ has a $\displaystyle j^2$ below it. Any help would be great. Thanks
    You can just get rid of the j. Suppose that $\displaystyle x_n$ is Cauchy, then
    $\displaystyle
    \sum\frac{|x_n(j)-x_m(j)|^2}{j^2} \rightarrow 0$

    In particular for each j
    $\displaystyle
    \frac{|x_n(j)-x_m(j)|^2}{j^2} \leq ||x_n-x_m|| \rightarrow 0$

    which can only happen if the top part of the LHS goes to zero (I am taking m,n to infinity).
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  3. #3
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    Thanks for the reply, okay so I was already assuming that $\displaystyle x_n$ was Cauchy, the bit I was confused about was the inequality

    $\displaystyle \frac{|x_n(j)-x_m(j)|}{j}\leq\|x_n-x_m\|<\epsilon$

    Which is need to show that $\displaystyle x_n(j)$ is also Cauchy, for each j.

    My question is: If you have the $\displaystyle \frac{1}{j}$ in the inequality, does that still satisfy the definition of Cauchy?
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  4. #4
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    Quote Originally Posted by ejgmath View Post
    Thanks for the reply, okay so I was already assuming that $\displaystyle x_n$ was Cauchy, the bit I was confused about was the inequality

    $\displaystyle \frac{|x_n(j)-x_m(j)|}{j}\leq\|x_n-x_m\|<\epsilon$

    Which is need to show that $\displaystyle x_n(j)$ is also Cauchy, for each j.

    My question is: If you have the $\displaystyle \frac{1}{j}$ in the inequality, does that still satisfy the definition of Cauchy?
    Why would it? You are considering j fixed, so if you really are worried about it just pick $\displaystyle \frac{\epsilon}{j}$.

    A formal way would be to say, fix j, and let epsilon be greater than zero, then there exists an N such that for all n,m>N;
    $\displaystyle
    \frac{|x_n(j)-x_m(j)|^2}{j^2} \leq ||x_n-x_m||< \frac{\epsilon^2}{j^2}$

    Thus for n,m>N
    $\displaystyle
    |x_n(j)-x_m(j)|< \epsilon$

    i.e. x_n(j) is Cauchy.
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