closure of polynomials in weak topoloy and weak* topology of $L^\infty(0,1)$.

Question

closure of polynomials in weak topoloy and weak* topology of $L^\infty(0,1)$?

I know $(L^1)'=L^\infty$, and the dual of $L^\infty$ is just Radon measures. How to find the closure of polynomials in weak topoloy and weak* topology of $L^\infty(0,1)$? I just know the basic definition in topology.