The a pproximation by Smooth functions and fixed point of semigroup with polyhedral norms on separable Banach spaces

Abstract

We characterize the separable Banach spaces . we also state sufficient conditions on aseparable Banach space so that the function can be taken to be of class . In particular we obtain the optimal order of smoothness of the approximation functions with no critical points on the classical spaces and . We show that every sub space of James tree space with non separable dual contains an isomorph of James tree complemented in James tree space . Also we show that every sub space of James function space with non separable dual contains an isomorphism of a new tree –like space . The new space yields that every sub space of James function with non-separable dual contains isomorphs of . We study fixed point properties of semigroups of non-expansive mappings on weakly compact convex sub sets of a Banach space . We provide a characterization for the existence of a left invariant mean on the space of weakly almost periodic functions on separable semitoplogical semigroups interms of fixed point property for non- expansive mappings . We present the existence of equivalent polyhedral norms on anumbers of clases of non- separable spaces .