Re: Another L^1 question....

Hello,

> Let f_n be a sequence of integrable functions on [0,1] such that f_n ---->
f
> a.e. and ||f_n||_2 <= 1 for n = 1, 2, 3, ...
> I am asked to :
> (i) show that f is in L^2[0,1]
> (ii) Show that ||f_n - f||_1 -----> 0 as n ----> infinity
>

(i) By Fatou's lemma, int(|f|^2) <= liminf int(|f_n|^2) <= 1, whence f is in
L^2[0,1].

(ii) Since f_n is in L^2[0,1], by Hölder inequality,
||f_n||_1 = int(|f_n|) <= int(|f_n|^2) <= 1.
Now the proof is identical to that of (i).

--
Julien Santini


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