What topology are we using in solving this problem?

Question

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?

Answer 1

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.