Uniformly Bounded Theorem And Fixed Point Theorem In B-banach Spaces

L~p[a,b]space is an important content of function space,through the continuous exploration and research of many scholars,they found that L~p[a,b](0<p<1)space has a very wide application in convex geometry analysis,optimizing CSP algorithm and LTPCSP algorithm.With the in-depth study of scholars at home and abroad,M.bota et al.Proposed a new class of normed linear spaces,namely b-normed linear spaces,based on the basic properties of spaces.In this paper,on the basis of the existing results,we consider the b-normed linear space deeply,and discuss the related properties of b-normed linear space,and according to these properties,uniform boundedness theorem,functional extension theorem of linear functional are proposed and proved.Then,based on the cone theory in b-normed linear space,the fixed point theorem of nonlinear operators in b-normed linear space is given.Finally,a class of special functions derived from normed linear space and b-normed linear space are proposed,namely normed preserving function and b-norm preserving function,and their properties and the relationship between them are discussed.