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Topological vector space with a locally convex topology, i.e. induced by a system of seminorms.

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Given projection PP is weak-star to weak-star continuous [closed]

Let $X$ be a Banach space, and $P$ be a projection of $X^*$ such that $\Vert P \Vert\leq 1$ and $\ker(P)$, ${\bf ran}(1-p)$ are weak-star closed. Show that $P$ is weak-star to weak-star continuous. …