### On Banach Spaces with the Gelfand-Philips Property.

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It is shown that if (S,∑,m) is an atomless finite measure space and X is a Banach space without the Radon-Nikodym property, then the quotient space cabv(∑,m;X)/L¹(m;X) is nonseparable.

The talk presented a survey of results most of which have been obtained over the last several years in collaboration with M.Florencio and P.J.Paúl (Seville). The results concern the question of barrelledness of locally convex spaces equipped with suitable Boolean algebras or rings of projections. They are particularly applicable to various spaces of measurable vector valued functions. Some of the results are provided with proofs that are much simpler than the original ones.

A simpler proof is given for the recent result of I. Labuda and the author that a series in the space L (lambda) is subseries convergent if each of its lacunary subseries converges.

For a map $\rho $ of $\mathbb{N}$ into itself, consider the induced transformation ${\sum}_{n}{x}_{n}\mapsto {\sum}_{n}{x}_{{\rho}_{(}n)}$ of series in a topological vector space. Then such properties of this transformation as sending convergent series to convergent series, or convergent series to bounded series, or bounded series to bounded series (and a few more) are mutually equivalent. Moreover, they are equivalent to an intrinsic property of ρ which reduces to those found by Agnew and Pleasants (in the case of permutations) and Wituła (in the general case) as...

For a relatively compact subset $S$ of the real line $$, let $R\left(S\right)$ denote the Banach space (under the sup norm) of all regulated scalar functions defined on $S$. The purpose of this paper is to study those closed subspaces of $R\left(S\right)$ that consist of functions that are left-continuous, right-continuous, continuous, and have a (two-sided) limit at each point of some specified disjoint subsets of $S$. In particular, some of these spaces are represented as $C\left(K\right)$ spaces for suitable, explicitly constructed, compact spaces...

CONTENTS1. Introduction..........................................................................................52. Vector measures with λ-everywhere infinite variation represented by series of simple measures.............113. Semicontinuity of some maps related to the variation map..................................................184. Sets of λ-continuous measures with (λ-) everywhere infinite variation.....................................235. Borel complexity of some spaces of vector measures........................................................266....

The area of research of this paper goes back to a 1930 result of H. Auerbach showing that a scalar series is (absolutely) convergent if all its zero-density subseries converge. A series ${\sum}_{n}{x}_{n}$ in a topological vector space X is called ℒ-convergent if each of its lacunary subseries ${\sum}_{k}{x}_{{n}_{k}}$ (i.e. those with ${n}_{k+1}-{n}_{k}\to \infty $) converges. The space X is said to have the Lacunary Convergence Property, or LCP, if every ℒ-convergent series in X is convergent; in fact, it is then subseries convergent. The Zero-Density Convergence...

Given an ordered metric space (in particular, a Banach lattice) E, the generalized Helly space H(E) is the set of all increasing functions from the interval [0,1] to E considered with the topology of pointwise convergence, and E is said to have property (λ) if each of these functions has only countably many points of discontinuity. The main objective of the paper is to study those ordered metric spaces C(K,E), where K is a compact space, that have property (λ). In doing so, the guiding idea comes...

The paper is devoted to a study of some aspects of the theory of (topological) Riesz space valued measures. The main topics considered are the following. First, the problem of existence (and, particularly, the so-called proper existence) of the modulus of an order bounded measure, and its relation to a similar problem for the induced integral operator. Second, the question of how properties of such a measure like countable additivity, exhaustivity or so-called absolute exhaustivity, or the properties...

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