Sequence, Lebesgue Space

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December 13th 2009, 01:48 PM
canberra1454

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?
December 13th 2009, 01:53 PM
Laurent

Why not apply the quoted result to $f_n=\frac{g_n^2}{n^2}$ ?... Try to find how to conclude from there.

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