Asymptotically Isometric Copies Of ι~β In β -normed Spaces

The theory of asymptotically isometric copy is one of the most active fields in functional analysis, and it has very important meanings for the fixed point property and other mathematic branches.In chapter 1, we take the research on theβ-normed space which contains anasymptotically isometric copy of l^β. We get characterizations of the family of all nonnegative, subadditive,β-absolutely homogeneous and continuous functionals defined on X , when theβ-normed space X contains an asymptotically isometric copy of l^β. Moreover, we have proved that if a closed boundedβ-convex subset B ofβ-normed spaces contains { xn } which is asym- ptotically isometric to l^β-basis, then B contains a closedβ-convex subset which fails the fixed piont property.In chapter 2, we study the asymptotically isometric copy of l^βin spaces of bounded linear operators for the first time. Combinating with the fixed point theory, we proved that assume X is a normed space and Y is aβ-normed space. If there is a quotient space of Y which is asymptotically isometric to l^β, then L ( X , Y ) contains an asymptotically isometric copy of l^β. Therefore L ( X , Y ) fails the fixed point property for nonexpansive mappings on closed boundedβ-convex subsets of L ( X , Y ).In chapter 3, we show the definition of a new operator called semi-bounded operator with semi-bounded sets in topological linear spaces. We proved that if E is a locally bounded or locally convex topological linear space and E₁ is a locally convex topological linear space, then T :Eâ†'E₁ is a bounded operator if and only if T is a semi-bounded operator. Moreover, if E with axiom A1 is a locally bounded or locally convex topological linear space, E₁ is a locally convex topological linear space, and 1T :Eâ†'E₁ is linear, then T is a continuous operator if and only if T is a semi-bounded operator.The last chapter introduces some kwoledge and problems about non-Archimedean fields and non-Archimedean normed spaces. We mainly study some results of foreign scholars', and propose some open problems.