Some Applications Of Ideal Convergence In Banach Spaces

Statistical convergence was first proposed by Zygmund in 1935,with almost one century development,its a mature system nowadays and has been applied in many branches of mathematics,in which Conor et al presented the equivalent characterizations of some properties of Banach space by statistical convergence,which inspired us to investigate locally compact sets in Banach space by I-convergence(I-convergence),one of the generalization of statistical convergence.we proved:for each non-empty closed convex set C in Banach space,each bounded sequence in C(weak)I-convergence if and only if C is locally(weakly)compact.On the other hand,we also investigate best proximinal.As an important part of proximinal theory,Rawashdek and others assumed that with condition" dim(F?G)<?",as F and G are respectively reflexive subspace and simultaneously proximinal in Banach space,we proved the algebra sum of F+G is proximinal.Based on this point.Let C and D are convex sets in Banach space,they are respectively weakly compact and simultaneously proximinal,we proved that C+Dis simultaneously proximinal.As a consequence without condition dim(F?G)<?,we proved that if F + Gis closed,then F + G is simultaneously proximinal.