We shall characterize the weak nearly uniform smoothness of the ψ-direct sum (X1 O…O XN)ψof N Banach spaces X1,..., XN, where ψ is a convex function satisfying certain conditions on the convex set [formula]. To do this, a class of convex functions which yield l1-like norms will be introduced. We shall apply our result to the fixed point property for nonexpansive mappings (FPP). In particular, an example which indicates that there are plenty of Banach spaces with FPP failing to be uniformly non-square will be presented.
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We shall characterize the weak nearly uniform smoothness of the fi-direct sum X Y of Banach spaces X and Y . The Schur and WORTH properties will be also characterized. As a consequence we shall see in the [...]-sums of Banach spaces there are many examples of Banach spaces with the fixed point property which are not uniformly non-square.
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We show that a separable Banach space X has the Schur property if and only if every separately compact bilinear application from X into c[sub 0] is completely continuous, thus answering a question raised by Pełczyński.
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