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Weak *-topology of $X^*$ is metrizable if and only if …
Define metrizability weak*-topology of $X^*$ which $X$ is T.V.S on which $X^*$ separates points.
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Ali Qurbani
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Definition of metrizability of X*, no X – Ali Qurbani May 20 '12 at 06:36
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I stated the definition for a general topological space $X$. Replace my $X$ by your $X^*$. – Robert Israel May 20 '12 at 06:45
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What you want is not completely clear. Do you want necessary/sufficient conditions on the space $X$ so that its dual $X^$ with the weak topology is metrizable? One well-known result is that the unit ball of $X^*$ is metrizable iff $X$ is separable, and the whole space is metrizable iff $X$ is finite-dimensional. – Najib Idrissi May 20 '12 at 08:31
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@N.I: that's not true. If $X$ has a countable Hamel basis, $X^*$ is metrizable. – Robert Israel May 20 '12 at 17:11
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@RobertIsrael: You're right, I was mistaken. – Najib Idrissi May 20 '12 at 17:58
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A topology on $X$ is metrizable if there is a metric on $X$ which produces the same topology.
Robert Israel
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