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If $X$ is an infinite dimensional separable Banach space (like $C^0(\Omega)$ ), and $\{x _ j \} _ {j\ge 1}$ is a dense sequence in its unit ball, one considers the norm on $X^*$ defined by $$ ||| u |||:=\sum _ {j=1} ^\infty 2 ^ {-j} |\langle u, x _ j \rangle |\\ ,$$ which is weaker than the dual norm $\|\cdot\|$, since $ |||u|||\le \| u\|$. It verifies: $u_j\to 0$ weakly$ ^*$ iff $|||u_j|||\to 0$, that is, the $|||\cdot|||$ norm topology and the $w^*$ topology have the same convergence structure, although they differ, because the latter is not metrizable. However, they induce the same topology on any bounded subset.