# Weak and weak* topologies

If X is a locally convex vectorspace, does the weak and weak* topologies on X* coinside? If so how to prove it?

• X is suppose to be a locally convex topological vector space. – 4Polynomials Apr 5 '16 at 9:50

No they do not coincide e.g. for $X=c_0$. Then $X^*=\ell^1$ and $X^{**}=\ell^\infty$. Then $\sigma(\ell^1,c_0)$ is strictly coarser than $\sigma(\ell^1,\ell^\infty)$ because $(\ell^1,\sigma(\ell^1,c_0))^*=c_0$ and $(\ell^1,\sigma(\ell^1,\ell^\infty))^*=\ell^\infty$.