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I was reading this question:

Show that $L^2[a,b]$ is of the first category

But I did not understand why the answer took $U := \{ f \in L^2[a,b]: \|f\|_{L^2[a,b]} \leq 1 \}$? could anyone explain this for me please?

Especially because in this link:

$L^2([0,1])$ is a set of first category in $L^1([0,1])$?

I saw a different set $B_{n}.$

  • The $B_n$ in the second post is the same as the $nU$ in the first post. – Thorgott Mar 18 '20 at 14:13
  • And why we want $U$ to take this form @Thorgott specifically? –  Mar 18 '20 at 14:54
  • The fundamental reason is that it accomplishes what we want it to accomplish. If you're asking for motivation for that choice of $U$, I'm afraid I can't help you, but then you should probably make that more explicit. – Thorgott Mar 18 '20 at 16:22
  • Yes I am asking about the intuition behind writing it write this @Thorgott –  Mar 19 '20 at 01:20

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