Why we take $U := \{ f \in L^2[a,b]: \|f\|_{L^2[a,b]} \leq 1 \}$?

Question

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 $B_{n}.$