The relationship between some smoothness and weak asymptotic-norming properties of dual Banach space X is studied. The main results are the following. Suppose that X is weakly sequential complete Banach space, then X is Frechét differentiable if and only if X has B(X)-ANP-I, X is quasi-Frechét differentiable if and only if X has B(X)-ANP-II and X is very smooth if and only if X has B(X)-ANP-II′. A new local asymptotic-norming property is also introduced, and the relationship among this one and other local asymptotic-norming properties and some topological properties is discussed. In addition, this paper gives a negative answer to the open question raised by Hu and Lin in Bull. Austral. Math. Soc,45,1992.
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