On Fermat-Torricelli Problem in Frechet Spaces

We study the Fermat-Torricelli problem (FTP) for Frechet space X, where X is considered as an inverse limit of projective system of Banach spaces. The FTP is defined by using fixed countable collection of continuous seminorms that defines the topology of X as gauges. For a finite set A in X consisting of n distinct and fixed points, the set of minimizers for the sum of distances from the points in A to a variable point is considered. In particular, for the case of collinear points in X, we prove the existence of the set of minimizers for FTP in X and for the case of non collinear points, existence and uniqueness of the set of minimizers are shown for reflexive space X as a result of strict convexity of the space.

λ-similar bases

Corresponding to an arbitrary sequence spaceλ, a sequence{xn}in a locally convex space (l.c.s.)(X,T)is said to beλ-similar to a sequence{yn}in another l.c.s.(Y,S)if for an arbitrary sequence{αn}of scalars,{αn p(xn)} ϵ λfor allp ϵ DT⇔{αn q(yn)} ϵ λ, for allq ϵ DS, whereDTandDSare respectively the family of allTandScontinuous seminorms generatingTandS.In this note we investigate conditions onλand the spaces(X,T)and(Y,S)which ultimately help to characterizeλsimilarity between two Schauder bases. We also determine relationship of this concept withλ-bases.

The countable neighbourhood property and tensor products

This article is intended to enlarge the study of spaces satisfying the countable neighbourhood property and to clarify the incidence of this property in the stability of some locally convex properties of tensor products.We shall use the standard notations of locally convex spaces as in [17] and [18]. The word space will always mean separated locally convex space. If (£, t) is a space, the set of all continuous seminorms on it will be denoted by cs(E). The linear hull and the absolutely convex hull of a subset C of a space will be written lin(C) and г(C) respectively.

Weakly Compact Sets in Orlicz Spaces

The purpose of this paper is to characterize weak compactness in Orlicz spaces. Though an Orlicz space is a Banach space, it will be viewed from the standpoint of the theory of Köthe spaces. Considering that a norm-bounded subset is not weakly compact in general, we shall give some criteria for weak compactness in terms of the functional defining an Orlicz space. Because weak compactness is closely connected with the continuity of the semi-norms on the conjugate space, at the same time some properties of continuous semi-norms on Orlicz spaces will be brought to light.The first characterization (Theorem 1) is concerned with degree of smoothness of the functional at the origin. In Theorem 2 a connection between weak compactness and boundedness (by another functional) is obtained. In Theorem 3 the result in Theorem 2 is stated as a proposition about continuous seminorms.