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 .
