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Here is the problem link (where the title is exactly what we want to prove)

$L^2$ is meager in $L^1$

My question is:

What topology are we using in solving this problem? could anyone explain this for me please?

1 Answers1

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The topology used is the one induced by the $1$-norm i.e. $\Vert f \Vert_1 := \int_{[0,1]} \vert f(x) \vert dx$.

This is well defined since $[0,1]$ has finite Lebesgue measure and thus $L^2([0,1]) \subseteq L^1([0,1])$.

That means, that the topology is generated by the family $\{B_{r}(f): r > 0, f \in L^1([0,1])\}$, where

$$B_{r}(f) := \{g \in L^1([0,1]) : \Vert f-g\Vert_1 < r\} ~~.$$

Hence a set $U \subseteq L^1([0,1])$ is open if and only if $U = \cup_{i \in I} B_{r_i}(f_i)$ where $I$ is some index set.

G. Chiusole
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  • and what is it explicitly? –  Mar 19 '20 at 10:48
  • what are the form of the open sets? –  Mar 19 '20 at 10:48
  • I have edited and added what that means explicitely. – G. Chiusole Mar 19 '20 at 10:52
  • why $f$ should belong to $L^1$ and not $L^2$? –  Mar 19 '20 at 11:02
  • You consider the set $L^2$ with the subspace topology as it is a subspace of $L^1$. The open sets of $L^2$ then have the form $U \cap L^2$ for any $U$ open in $L^1$. – G. Chiusole Mar 19 '20 at 11:03
  • One more question please: why this set that is defined in a link contained in the above link : For $n \in \mathbb N$ set $B_n = {f \in L^2[0,1]\mid |f|_2 \le n }$ has $\leq n$ and not <n only? –  Mar 19 '20 at 11:06
  • And why we are taking in the set I mentioned in my last comment $f \in L^2$ and not $L^1$? –  Mar 19 '20 at 11:08
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    What we want to show is that $L^2$ is meager i.e. that it is the union of nowhere dense sets. So if we can show that $L^2 \subseteq \cup {f \in L^1: \Vert f \Vert_1 \leq n }$, or that $L^2 \subseteq \cup {f \in L^2: \Vert f \Vert_1 \leq n }$, or that $L^2 \subseteq \cup {f \in L^2: \Vert f \Vert_1< n }$ or $L^2 \subseteq \cup {f \in L^2: \int \vert f\vert^2 dx < n }$. Any of those will suffice. – G. Chiusole Mar 19 '20 at 11:17
  • ohh why any of those sets are the same? –  Mar 19 '20 at 11:29
  • why the last set contains the square root of $|.|_{2}$? –  Mar 19 '20 at 11:31
  • I will post now a new question about my last 2 comments I hope you can answer it for me …. I will provide you with the link here. –  Mar 19 '20 at 11:32
  • Also, if you have time could you look at this question for me please? https://math.stackexchange.com/questions/3585432/why-we-take-u-f-in-l2a-b-f-l2a-b-leq-1 (I am still typing the other question) –  Mar 19 '20 at 11:37
  • Here is the link https://math.stackexchange.com/questions/3586693/a-question-about-the-sets-used-in-the-proof-of-l2-is-meager-in-l1 –  Mar 19 '20 at 11:56