The Finite-dimensional Decomposition Property in Non-Archimedean Banach Spaces  

作　　者：Albert KUBZDELA Cristina PEREZ-GARCIA 

机构地区：[1]Institute of Civil Engineering,Poznań University of Technology,Ul.Piotrowo 5,61-138 Poznań,Poland [2]Department of Mathematics,Facultad de Ciencias,Universidad de Cantabria,Avda.de los Castros s/n,39071,Santander,Spain

出　　处：《Acta Mathematica Sinica,English Series》2014年第11期1833-1845,共13页数学学报（英文版）

基　　金：partially supported by Ministerio de Ciencia e Innovación,MTM2010-20190-C02-02

摘　　要：A non-Archimedean Banach space has the orthogonal finite-dimensional decomposition property（OFDDP） if it is the orthogonal direct sum of a sequence of finite-dimensional subspaces.This property has an influence in the non-Archimedean Grothendieck＇s approximation theory,where an open problem is the following： Let E be a non-Archimedean Banach space of countable type with the OFDDP and let D be a closed subspace of E.Does D have the OFDDP？ In this paper we give a negative answer to this question; we construct a Banach space of countable type with the OFDDP having a one-codimensional subspace without the OFDDP.Next we prove that,however,for certain classes of Banach spaces of countable type,the OFDDP is preserved by taking finite-codimensional subspaces.A non-Archimedean Banach space has the orthogonal finite-dimensional decomposition property（OFDDP） if it is the orthogonal direct sum of a sequence of finite-dimensional subspaces.This property has an influence in the non-Archimedean Grothendieck＇s approximation theory,where an open problem is the following： Let E be a non-Archimedean Banach space of countable type with the OFDDP and let D be a closed subspace of E.Does D have the OFDDP？ In this paper we give a negative answer to this question; we construct a Banach space of countable type with the OFDDP having a one-codimensional subspace without the OFDDP.Next we prove that,however,for certain classes of Banach spaces of countable type,the OFDDP is preserved by taking finite-codimensional subspaces.

关 键 词：Non-Archimedean Banach spaces finite-dimensional decomposition property orthogonal base 

分 类 号：O177.2[理学—数学] O1-09[理学—基础数学]