In the next problem of Rudin's book, Functional Analysis,
Let $L^1$ and $L^3$ be the usual Lebesgue spaces on the unit interval. Prove that $L^2$ is of the first category in $L^1$, in the following way
- Put $g_n = n$ on $[0, n^{-3}]$, and show that $$ \int fg_n \to 0 $$ for every $f \in L^2$, but not for every $f \in L^1$.
I proved the statement, but I'm having problems to prove this implies that $L^2$ is of the first category in $L^1$. I think that I can use Banach-Steinhaus theorem, taking $\{\int fg_n \}$ as a collection of linear mappings of $L^1$ to $\mathbb{C}$, but not sure of this.
Any suggestions?