Is there some reason that the fact that $L^\infty$ is the dual space of $L^1$ is an important fact?

Question

Why is the fact that $L^\infty$ is the dual space of $L^1$ an important fact?

Answer 1

For a sigma-finite measure $\mu$, $L^\infty(\mu)$ is the dual of $L^1(\mu)$. However, it is not the case that $L^1$ is also the dual of $L^\infty$! See this and this for more details.

Answer 2

The fact that $L^\infty$ is the dual space of $L^1$ implies that its closed unit ball is compact with respect to the weak-* topology. This is an important property of $L^\infty$.

The notion of weak-* convergence in $L^\infty$ can be illustrated by the Riemann-Lebesgue lemma that states that, for all $f \in L^1({\bf R})$, we have the convergence $$ \int_{\bf R} f(t) e^{int} dt \rightarrow 0 \hbox{ as } n \rightarrow \infty. $$ This is the convergence of the sequence $\{e^{int}\}_n$ to zero in the weak-* topology of $L^\infty$. The Riemann-Lebesgue lemma is certainly an important result in Fourier analysis and the compactness of the unit ball of $L^\infty$ may be used for example to prove generalizations of the Riemann-Lebesgue lemma.