Recall that a functor F is continuous is the map from Hom(V，W）to Hom(F(V),F(W)) is always continuous. I have already know how to prove the functor V** is continuous, but don't know why the functor F(V)=V* is continuous. Please give some advice about that.
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Hint: Fix a basis, and ask your self what the entries of $T^\ast$ are in terms of the entries of $T$. It should be pretty obvious then. – Alex Youcis Oct 04 '13 at 04:11
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Ok, I see. Thanks. – Jack Oct 04 '13 at 04:24