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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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    Definition of metrizability of X*, no X2012-05-20
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    I stated the definition for a general topological space $X$. Replace my $X$ by your $X^*$.2012-05-20
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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.2012-05-20
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    @N.I: that's not true. If $X$ has a countable Hamel basis, $X^*$ is metrizable.2012-05-20
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    @RobertIsrael: You're right, I was mistaken.2012-05-20

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A topology on $X$ is metrizable if there is a metric on $X$ which produces the same topology.