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Weak-to-weak continuous operator which is not norm-continuous

I saw in some text that if an operator $T:Y^\star \rightarrow X^\star$ (where $X$ and $Y$ are separable Banach spaces) is $w^\star - w^\star$ - continuous then $T$ is the adjoint of some operator $S:X\rightarrow Y$. Can anyone write a short proof for this claim or give some reference?

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    @Martin: Anybody searching for this title will find this question, which has a perfectly good link to the previous one.2012-11-07

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