Some Pointwise Properties Of Banach Spaces And Metric Selection Of Linear Relation

This dissertation concerns with the heritage of the Banach space X i's pointwise properties from Lebesgue-Bochner sequence spaces l p ( X i), and the relationships between some well known pointwise properties in the Banach space. The contents of the dissertation can be divided into three parts:First, it traces the development of the theories of Lebesgue-Bochner sequence spaces l p ( X i) and his function spaces L p(μ, X), summarizes the achievements made by many scholars in the related field, and describes the background and significance of the Lebesgue-Bochner spaces.Second, this dissertation argues the heritage of H point, WM point, power U point, WLUR point, CLUR point, and WCLUR point in the Lebesgue-Bochner sequence spaces l p ( X i) from H point, WM point, power U point, WLUR point , CLUR point, and WCLUR point in the Banach space X i for each i∈N in great details in Chapters II, III and IV. In the three Chapters, we prove that the unit ball S ( X ) of Lebesgue-Bochner sequence space l p ( X i) has H point, WM point, power U point, WLUR point, CLUR point, and WCLUR point properties if and only if the unit ball S ( X ) of the Banach space X i has H point, WM point , power U point, WLUR point, CLUR point, and WCLUR point properties for each i∈N,then we lift the point properties of the Banach spaces X i for each i∈N to the Lebesgue-Bochner sequence spaces l p ( X i).Third, this dissertation argues various relations among the Banach spaces. In this section we prove that for each x∈S ( X) of Banach space X is CLUR point if and only if x∈S ( X) is WCLUR point, WM point and H point, and the following we prove that for each x∈S ( X) of Banach space X is WLUR point if and only if x∈S ( X) is WCLUR point , WM point and power U point.At last, based on Banach spaces theory, we will obtain important results by through investigating the metric Selection of linear Relation.