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Math Help - Limit of a Sequence of Elements in L^2

  1. #1
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    Limit of a Sequence of Elements in L^2

    I am asked to prove: If $\displaystyle \lim_{n \rightarrow \infty} ||f_{n} - f|| = 0$ then $\displaystyle \lim_{n \rightarrow \infty} ||f_{n}|| = ||f||$. I've been trying to answer this by an epsilon-delta argument, but I wonder if this is the best way. Should I try to do some algebra, converting ||f_{n} - f|| into an integral and then try to push the limit inside? If so, I'm not certain what permits me to move the limit inside a root.
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  2. #2
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    Progress on the epsilon-delta argument:

    $\displaystyle \lim_{n \rightarrow \infty} ||f_{n} - f|| = \lim_{n \rightarrow \infty}\Bigl( \int_{I} f_{n}^{2}(x) - 2f_{n}(x)f(x) + f^{2}(x) \Bigr)^{\frac{1}{2}}$. Let $\varepsilon > 0$ be given, then there exists an $N \in \mathbb{N}$ such that $\displaystyle n \geq N \Rightarrow \Bigl( \int_{I} f_{n}^{2}(x) - 2f_{n}(x)f(x) + f^{2}(x) \Bigr)^{\frac{1}{2}} < \varepsilon \Rightarrow \int_{I} f^{2}_{n}(s) - 2\int_{I}f_{n}(x)f(x) + \int_{I} f^{2}(x) < \varepsilon^{2}$. On the other hand, $\displaystyle \lim_{n \rightarrow \infty} ||f_{n}|| = ||f|| \Leftarrow ||f_{n}|| - ||f||  = \Bigl| \Bigl( \int_{I} f_{n}^{2} \Bigr)^{\frac{1}{2}} -  \Bigl( \int_{I} f^{2} \Bigr)^{\frac{1}{2}} \Bigr| < \varepsilon \Leftarrow \int_{I} f^{2}(x) - 2 \Bigl( \int_{I} f_{n}(x)f(x) \Bigr)^{\frac{1}{2}} + \int_{I} f^{2} < \varepsilon^{2}$. It seems that all I have to do is deal with the square root, but since the quantities may be greater or less than 1, I'm not sure how to proceed.
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  3. #3
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    ||\cdot|| is a norm, so you can use the triangle inequality :
    ||f_{n}-f||\geq|||f_{n}||-||f|||
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  4. #4
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    I think it's like this:

    [LaTeX ERROR: Convert failed]

    From the first norm you have [LaTeX ERROR: Convert failed]

    [LaTeX ERROR: Convert failed]

    Substituting this into the top thing we get:

    [LaTeX ERROR: Convert failed] .

    We gotta show [LaTeX ERROR: Convert failed] ?

    I think it'll go along the lines of:

    [LaTeX ERROR: Convert failed]

    However, if you use what iondor posted, it becomes much easier.
    Last edited by Vlasev; August 1st 2010 at 12:04 PM.
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