If X is a locally convex vectorspace, does the weak and weak* topologies on X* coinside? If so how to prove it?
2026-03-27 01:13:35.1774574015
Weak and weak* topologies
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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 $(\ell^1,\sigma(\ell^1,\ell^\infty))^*=\ell^\infty$.