weak topology and weak* topology on $L^1, L^{\infty}$

Question

Suppose $L^1(I)$ is the primal space and $L^{\infty}(I)$ is the dual. Could I simultaneously define weak topology on $L^1(I)$ with respect to $L^{\infty}(I)$ and define weak or weak* topology on $L^{\infty}(I)$ with respect to $L^1(I)$ and use these definitions simultaneously?

Answer 1

The topologies induced by the dual pairing $\sigma:L^1\times L^\infty \to \mathbb R$, defined by $$ \sigma(u,v) = \int_I uv\ dx $$ make them topological vector spaces with weak and weak-star topology:

$L^1$ supplied with topology induced by seminorms $p_v(u):=|\sigma(u,v)|$ coincides with $L^1$ supplied with the weak topology.

This follows from the fact that the mapping $v\mapsto f_v$, $f_v(u)=\int_I uv \ dx$, is an isometry between $L^\infty(I)$ and $L^1(I)^*$.