Is the union of all $l^p$ spaces meagre in $l^\infty$? i.e. is $ \bigcup_{p=1}^\infty l^p$ meagre?

I am revisiting this variety of math after a long break so help is appreciated. Please correct anything I get wrong too.

I know that a space is meagre if it can be written as a countable union of nowhere dense sets, and that a set is nowhere dense if it is closed and has no interior points.

My attempts/ideas:

(1) I seem to remember that $l^1 \subset l^2 \subset l^3 \subset l^4 \subset ...\subset l^\infty$, but maybe I'm wrong about this? I can't seem to find it anywhere or prove it either. Is this true? If this were the case then it would be enough to show that $\bigcup_{p=1}^\infty l^p=l^\infty$ is meagre. Then I'm stuck again... how do I show that $l^\infty$ is meagre?

(2) There is already a countable union (of $l^p$s) here (do I have to show that it is countable? This is obvious isn't it?), so that makes me think that it would be good if I could show that each $l^p$ is nowhere dense. How could I do this?

Any critiques/ideas?

  • $\begingroup$ It seems to me you would need to specify the space in which you want to show that the union is meagre. Also, the union is contained in $\ell^\infty$, equality does not hold! Your comment (1) about the inclusion is true. In order to see why, note that $|x_n|<1$ for all but finitely many $n's$. This allows you to go from the a priori uncountable union of $p\in[1,\infty)$ to the countable union of $p\in\mathbb N$. Also, you may find math.stackexchange.com/questions/942706/… interesting. $\endgroup$ – Jonas Dahlbæk Sep 23 '14 at 10:53
  • $\begingroup$ @user161825 Sorry, I mean to show the union is meagre in $l^\infty$. Edited $\endgroup$ – JaSoNmAtHgUy Sep 23 '14 at 10:57
  • $\begingroup$ @user161825 That's my question too! Will that one help me here too? I don't see how to apply it. Also, why doesn't the equality hold? You say that the union is contained within $l^\infty$ but isn't $l^\infty$ also contained in the union? $\endgroup$ – JaSoNmAtHgUy Sep 23 '14 at 11:01
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    $\begingroup$ Hint: The union of all the $\ell_p$ spaces is contained in the space $c_0$ of all sequences that converge to $0$. $\endgroup$ – Mathmo123 Sep 23 '14 at 11:06
  • $\begingroup$ @Mathmo123 Why is that true? I'm confused. $\endgroup$ – JaSoNmAtHgUy Sep 23 '14 at 11:18

Observe that if $$x\in \bigcup_{p\in[1,\infty)}\ell^p,$$ then $x\in \ell^p$ for some $1\leq p <\infty$. thus $\sum |x_n|^p<\infty$. In particular $|x_n|\rightarrow 0$, so $$\bigcup_{p\in[1,\infty)}\ell^p\subset c_0.$$

  • $\begingroup$ Why does $|x_n| \to 0$? $\endgroup$ – JaSoNmAtHgUy Sep 23 '14 at 11:17
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    $\begingroup$ What happens if it doesn't? Can the series then converge? $\endgroup$ – Mathmo123 Sep 23 '14 at 11:19
  • $\begingroup$ @Mathmo123 Oh ok, I understand why. So the union is a subspace of $c_0$. Then I want to show that the union is closed in $c_0$ and has no interior points? Implying that it is meagre. Is that right? $\endgroup$ – JaSoNmAtHgUy Sep 23 '14 at 11:28
  • $\begingroup$ Rather, you want to argue that any subset of a meagre set is meagre. $\endgroup$ – Jonas Dahlbæk Sep 23 '14 at 11:29
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    $\begingroup$ @user161825 Ok, ambiguous notation is what killed my efforts before I posted this question then. Thanks for all your help. $\endgroup$ – JaSoNmAtHgUy Sep 23 '14 at 12:08

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