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Math Help - L^p spaces

  1. #1
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    L^p spaces

    Let p>1. Give examples of sequences (f_n) and (g_n) in L^{p} (0,1) such that

    i) lim_{n\rightarrow \infty} f_n(x) = 0 almost everywhere but lim_{n\rightarrow \infty} ||f_n||_{p} \neq 0 .

    ii) lim_{n\rightarrow \infty} ||g_n||_{p} = 0 but lim_{n\rightarrow \infty} g_n(x) does not exist for any x \in (0,1).
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  2. #2
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    Quote Originally Posted by HenryB View Post
    Let p>1. Give examples of sequences (f_n) and (g_n) in L^{p} (0,1) such that

    i) lim_{n\rightarrow \infty} f_n(x) = 0 almost everywhere but lim_{n\rightarrow \infty} ||f_n||_{p} \neq 0 .

    ii) lim_{n\rightarrow \infty} ||g_n||_{p} = 0 but lim_{n\rightarrow \infty} g_n(x) does not exist for any x \in (0,1).
    For i), try f_n(x) = n^{1/p}x^n.

    ii) is trickier. You need to use something like the Haar functions.
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  3. #3
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    Thanks a lot. (i) is a good answer but is there really nothing simpler for (ii)? It doesn't seem like the sort of solution that you can really spot yourself.

    The question goes on to say:

    "For each \epsilon > 0 fnd a subset E_\epsilon of [0,1] such that f_n(x) \rightarrow 0 uniformly on [0,1] \ E_\epsilon.

    Now it's clearly not what the question is after (I don't actually know what the question is trying to say) but can't we just take E_\epsilon = (0,1] for all \epsilon > 0?

    Secondly, can we find a subsequence ({g_n}_r) such that lim_{r \rightarrow \infty} {g_n}_r(x) = 0 almost everywhere?
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  4. #4
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    Quote Originally Posted by HenryB View Post
    is there really nothing simpler for (ii)? It doesn't seem like the sort of solution that you can really spot yourself.
    The difficulty with (ii) is that to ensure \|g_n\|_p\to0 you want g_n to be small over most of the interval; but to prevent \lim_{n\to\infty}g_n(x) existing at every point of the interval you need to arrange that, for every x, g_n(x) stays well away from 0 for infinitely many values of n. The only convenient way I know to achieve that is to define g_n to be 0 on most of the interval, but to bob up to 1 on a small subinterval that gradually works its way along the whole interval.

    The easiest way to do this is to construct g_n for n in batches of length 2^k. For n = 2^k+j, where 0\leqslant j<2^k, define g_n(x) to be 1 on the interval \left[\frac j{2^k},\frac{j+1}{2^k}\right] , and 0 on the remainder of the interval [0,1].

    Quote Originally Posted by HenryB View Post
    The question goes on to say:

    "For each \epsilon > 0 find a subset E_\epsilon of [0,1] such that f_n(x) \rightarrow 0 uniformly on [0,1] \ E_\epsilon.

    Now it's clearly not what the question is after (I don't actually know what the question is trying to say) but can't we just take E_\epsilon = (0,1] for all \epsilon > 0?

    Secondly, can we find a subsequence ({g_n}_r) such that lim_{r \rightarrow \infty} {g_n}_r(x) = 0 almost everywhere?
    For the example f_n(x) = n^{1/p}x^n that I gave, the only point at which f_n(x)\not\to0 is x=1. The sequence does not converge uniformly on [0,1], or even on [0,1), but it does converge to 0 uniformly on [0,1-\varepsilon], for all \varepsilon > 0.

    For the above sequence (g_n), the subsequence (g_{2^k}) converges to 0 almost everywhere (in fact, everywhere except at x=0).
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  5. #5
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    Great help - thanks a lot indeed.
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