In a normed linear space X, consider a nonempty closed set K which has the property that for some r > 0 there exists a set of points xo € X\K, d(xoK) > r, which have closest points p(xo) € K and where the set of points xo — r((xo — p(xo))/\\xo — p(zo)||) is dense in X\K. If the norm has sufficiently strong differentiability properties, then the distance function d generated by K has similar dif...

Let K(X,Y) be the space of compact operators. For a proximinal subspace Z⊂Y, this paper deals with question, when does every Y-valued operator admit Z-valued best approximation? any reflexive Banach X and for L1-predual Y, if Z⊂Y is strongly finite codimension, we show that K(X,Z) under an additional condition on position K(X,Z). When Y c0-direct sum dimensional spaces achieve strong transitivi...

Let X be a Banach space, C a closed subset of X , and T : C -+ C a nonexpansive mapping. Conditions are given which assure that if the fixed point set F ( T ) of T has nonempty interior then the Picard iterates of the mapping T always converge to a point of F ( T ) . If T is asymptotically regular, it suffices to assume that the closed subsets of X are densely proximinal and that nested spheres...

Let (S,Σ , μ) be a complete positive σ -finite measure space and let X be a Banach space. We are concerned with the proximinality problem for the best simultaneous approximations to two functions in L p(S,Σ , X). Let Σ0 be a sub-σ -algebra of Σ and Y a nonempty locally weakly compact convex subset of X such that span Y and its dual have the Radon–Nikodym property. We prove that L p(S,Σ0, Y ) is...

we study the theory of best approximation in tensor product and the direct sum of some lattice normed spacesx_{i}. we introduce quasi tensor product space anddiscuss about the relation between tensor product space and thisnew space which we denote it by x boxtimesy. we investigate best approximation in direct sum of lattice normed spaces by elements which are not necessarily downwardor upward a...

in this paper we obtain a necessary and a sufficient condition for the set of ω_0-nearest points ( ω_0-farthest points) to be non-empty or a singleton set in normed linear spaces. we shall find a necessary and a sufficient condition for an uniquely remotal set to be a singleton set.

S OF TALKS (the information in brackets refers to session numbers) On Simultaneous Approximation in Function Spaces [C-5B] E. Abu-Sirhan Tafila Technical University, Tafila, Jordan [email protected] The problem of simultaneous approximation in function spaces has attracted many researcher recently. Major results on the space of vector valued continuos functions started to appear early nineties...

We study the theory of best approximation in tensor product and the direct sum of some lattice normed spacesX_{i}. We introduce quasi tensor product space anddiscuss about the relation between tensor product space and thisnew space which we denote it by X boxtimesY. We investigate best approximation in direct sum of lattice normed spaces by elements which are not necessarily downwardor upward a...