The Research Of Binomial Difference Sequence Spaces And N-normed Sequence Spaces

Sequence space theory is an important branch of functional analysis.The research of sequence space and its operators is widely applied in the fields of summability,complex functions and so on.The aim of this paper is to investigate some definitions and properties of binomial difference sequence spaces and n-normed space valued sequence spaces.In the first chapter,the background and significance of the study are described,and the obtained results of sequence spaces are specifically stated in Euler difference sequence spaces and n-normed spaces valued sequence spaces.In the second chapter,the definitions of the binomial difference sequence spaces b0r,s(?),bcr,s(?),b?r,s(?),b0r,s(?(m)),bcr,s(?(m)),b?(?(m)),b0a,b(B(m)),bca,b(B(m),b?a,b(B(m))are given by considering the definitions of the binomial matrix and the difference operators.First,the strict inclusion relation between the binomial difference sequence spaces and the Euler difference sequence spaces is proved,which implies that the significance of the binomial difference sequence space exists.Moreover,in order to discuss the infinite matrix transformation of sequence space conveniently,we prove the BK-property of the binomial difference sequence spaces,and compute Schauder bases of some of these sequence spaces.Finally,in order to study linear operators of binomial difference sequence spaces conveniently,?-,?-,?-duals of the binomial difference sequence spaces are obtained.In the third chapter,first,according to the definition of n-normed space valued sequence spaces,some conditions of equivalence of convergence are proved in natural n-normed space lp-valued sequence spaces.Moreover,the aim is to study the existence of n-norm and derived norms on n-normed space valued sequence spaces Z(Xk).And some results on equivalence of convergence in different norms are proved under certain cases.Using these results,the fixed point theorem is proved in the n-normed space Z(Xk).