Results 1 to 2 of 2

Math Help - Sequence, Lebesgue Space

  1. #1
    Newbie
    Joined
    Mar 2009
    Posts
    6

    Sequence, Lebesgue Space

    Prove that if (g_n) is a sequence in L^2[0, 1] with || g_n ||_2 \leq 1 for all n \geq 1 then (g_n/n) converges to zero a.e.


    This should follow from this:

    If (f_n)_n is a sequence in L^1[0, 1] is such that \sum_{n=1}^{\infty}  || f_n ||_1 < \infty then \sum_{n=1}^{\infty} | f_n(s) | < \infty for almost every s \in [0, 1].


    However, I do not see how this would follow. How do I prove this?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2008
    From
    Paris, France
    Posts
    1,174
    Why not apply the quoted result to f_n=\frac{g_n^2}{n^2}?... Try to find how to conclude from there.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Lebesgue measure of a manifold embedded in the euclidean space
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: November 30th 2011, 07:35 PM
  2. Sequence, Lebesgue Space
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: December 15th 2009, 10:46 AM
  3. Sequence in Lebesgue Space
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: December 13th 2009, 01:48 PM
  4. Question about lebesgue space and fourier transforms
    Posted in the Advanced Math Topics Forum
    Replies: 4
    Last Post: December 4th 2008, 10:19 PM
  5. Sequence of Lebesgue measurable functions
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 14th 2008, 11:51 PM

Search Tags


/mathhelpforum @mathhelpforum