A friend of mine found a book in which the author said that the dual space of $L^\infty$ is $L^1$, of course not with the norm topology but with the weak-* topology. Does anyone know where I can find this result? Thanks.
For any $C(K)$-space we have $C(K)^*\cong L_1(\mu)$ for some usually humongous measure $\mu$. See the proof of Proposition 4.3.8(iii) in
F. Albiac, N.J. Kalton, Topics in Banach Space Theory, Grad. Texts in Math. 233, Springer, 2006.
Of course, $L_\infty(\nu)\cong C(K)$ for some compact, Hausdorff space $K$. However, there is no clear relation between the measures $\mu$ and $\nu$. In fact, if $L_\infty(\nu)$ is infinite-dimensional, then $\mu$ is not even $\sigma$-finite.
There is a general fact from duality of linear spaces (see Proposition 4.28 in Fabian-Habala-Hajek-Montesinos-Pelant-Zizler, Functional Analysis and Infinite-Dimensional Geometry): If we consider a linear subspace $F$ in the space of linear functionals on $E$, then the space of linear functionals on $E$ continuous in the corresponding weak topology on $E$ coincides with $F$.