In this paper, we give four general results on linear extension of isometries between the unit spheres in β-normed spaces. These results improve the corresponding theorems in β-normed spaces.
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 Frechet differentiable if and only if X has B （X）- ANP -I, X is quasi-Frechet differentiable if and only if X has B（X）- ANP -H 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.
The sufficient and necessary conditions of the Bochner-Orlicz sequence spaces equipped with the Luxemburg norm, which have λ property, are given. The result shows, which is not as usual, that the λ property of the Bochner-Orlicz sequence spaces can not be lift from X.
