every weakly compact set K in locally convex space E is metrizable.
this is tru?
if not please give me an example.
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You probably ask if $K$ is metrizable in the weak topology. – Jun 08 '17 at 10:29
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yes. every weakly compact set K in locally convex space E is metrizable? – jafar Jun 08 '17 at 10:31
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No.
Let $X$ be a reflexive non-separable Banach space and denote by $X_{\leq 1}$ the closed unit ball of $X$.
By Banach-Alaoglu we know that $X_{\leq 1}$ is weakly compact. If $X_{\leq 1}$ is metrizable, then $X$ must be separable.
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thank you. for locally convex space with countable semi norm it is true? – jafar Jun 08 '17 at 10:40