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$L^1$, $L^2$: usual Lebesgue spaces on the unit interval. Show that $L^2$ is of the first category (meager) in $L^1$, in three ways:

(a) Show that $F_n:=\{f:\int|f|^2 \leq n\}$ is closed in $L^1$ but has empty interior.

(b)Put $g_n=n$ on $[0,n^{-3}]$, and show that $\int fg_n \rightarrow 0$ $\forall f \in L^2$ , but not for every $f\in L^1$.

(c) Note that the inclusion map of $L^2$ into $L^1$ is continuous but not onto.

Do the same for $L^p$ and $L^q$ if $p<q$.

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What have you tried? Copying an exercise verbatim from a book, with no thoughts of your own is not very helpful. –  mrf Jan 8 '13 at 21:08
    
@mrf, I have tried to solved this problem. Can you give me some idea? –  user52523 Jan 8 '13 at 21:14

1 Answer 1

For each question, there are two sub-questions: 1. Why the assertion is true? and 2. Why does this one give the result?

a. 1. Show sequential closeness (as we are in a metric context) by Fatou's lemma. What happens if we assume that $F_n$ has a non-empty interior (in $L^1$)? 2 It's by definition.

b. 1. Use the fact that $\{f_n\}$ is bounded in $L^2$ and that simple functions are dense in $L^2$.

c. 1. It follows from a well-known integral inequality. Take an integrable function which is not square integrable. 2. There is a related theorem earlier in the book.

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Thanks for susgesting ideas. I will try this problem. –  user52523 Jan 8 '13 at 21:40
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@user52523 Any progress? –  Davide Giraudo Jan 10 '13 at 18:13
    
I have some tried but i still have no idea for a, and c. With b, i solve it in the following way: b, $\int |fg_n|\le\alpha\int|g_n|$, but we have: $\int g_n=n.1/n^3 \rightarrow 0, $ so is $\int fg_n$ Could you check out for me, please. Thanks you so much. –  user52523 Jan 11 '13 at 12:51
    
I just finished this problem. Thank you so much for helping me. –  user52523 Jan 13 '13 at 19:55

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