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Sequentially continuous but not continuous linear map $(X^*, w^*)$ to $(Y^*,w^*)$

Let $X, Y$ be Banach spaces and let $T : (X^*, w^*) \rightarrow (Y^*,w^*)$ be a linear map. Suppose that $T$ is sequentially continuous. Must $T$ be weak-to-weak-continuous ?

M.G
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