We mean here $L_2$, and $L_1$ the usual Lebesgue spaces on the unit-interval. It is excercise 2.4 from Rudin. There's several ways to show that $L_2$ is nowhere dense in $L_1$.
But in (b) they ask to show that
$$\Lambda_n(f)=\int fg_n \to 0 $$
where $g_n = n$ on $[0,n^{-3}]$ and 0 otherwise, holds for $L_2$ but not for all $L_1$. Apparantly this implies that $L_2$ is of the first Category, but I dont know how. Second, I can show this holds for $L_2$ but I cant find a counterexample in $L_1$.
Theorem 2.7 in Rudin says:
Let $\Lambda_n:X\to Y$ a sequence of continuous linear mappings ($X,Y$ topological vector spaces)
If $C$ is the set of all $x\in X$ for which $\{\Lambda_n x\}$ is Cauchy in $Y$, and if $C$ is of the second Category, then $C=X$.
So if we find a $f\in L_1$ such that $\Lambda_n(f)$ is not Cauchy, then we proved that $L_2\subset C \subset L_1$ is of the first category. However I dont see why showing that $\Lambda_n(f)$ does not converge to 0 for some $f\in L_1$ is enough here.
Am I missing something?
The simplest functions in $L_1 \setminus L_2$ are $f_\alpha \colon x \mapsto x^\alpha$ with $-1 < \alpha \leqslant -\frac12$.
Computing $\int fg_n$ for such an $f_\alpha$ yields
$$\begin{align} \int f_\alpha g_n &= n\int_0^{n^{-3}} x^\alpha\,dx \\ &= \frac{n}{1+\alpha}n^{-3(1+\alpha)}\\ &= \frac{n^{-2-3\alpha}}{1+\alpha}. \end{align}$$
We see that the sequence of integrals does not converge to $0$ iff $-2-3\alpha \geqslant 0 \iff \alpha \leqslant -\frac23$.
Regarding the second part, choosing $-1 < \alpha < -\frac23$ gives an $f\in L_1$ with $\Lambda_n(f) \to \infty$, so $\Lambda_n(f)$ certainly is not a Cauchy sequence. Choosing $\alpha = -\frac23$ gives an $f\in L_1$ such that $\Lambda_n(f)$ is constant, hence a Cauchy sequence, but does not converge to $0$.
Now, if $L_2$ were of the second category in $L_1$, then the fact that $\Lambda_n(f) \to 0$ for all $f\in L_2$ would imply that $\Lambda_n(f) \to 0$ for all $f\in L_1$, by part $(b)$ of theorem 2.7. But picking $\alpha < -\frac23$ to get an $f\in L_1$ such that $\Lambda_n(f)$ is not a Cauchy sequence seems preferable, since it's more direct.