We characterize Banach lattices on which every positive almost Dunford-Pettis operator is weakly compact.

The $\sigma$-property of a Riesz space (real vector lattice) $B$ is: For each sequence $\left\{b_n\right\}$ of positive elements of $B$, there is a sequence $\left\{{\lambda }_n\right\}$ of positive reals, and $b\in B$, with ${\lambda }_n{b}_n\le b$ for each $n$. This condition is involved in studies in Riesz spaces of abstract Egoroff-type theorems, and of the countable lifting property. Here, we examine when “$\sigma$” obtains for a Riesz space of continuous real-valued functions $C\left(X\right)$. A basic result is: For discrete $X$, $C\left(X\right)$ has $\sigma$ iff the cardinal $\left|X\right|<𝔟$, Rothberger’s bounding number. Consequences and...

In this paper we introduce the notion of topological G-spaces. This is an intermediate class between G-sets and A-modules. After giving suitable definitions and illustrating examples, we prove a theorem of closed graph theorem type.

Sufficient conditions are given in order that, for a bounded closed convex subset $B$ of a locally convex space $E$, the set $C(X,B)$ of continuous functions from the compact space $X$ into $B$, is the uniformly closed convex hull in $C(X,E)$ of its extreme points. Applications are made to the unit ball of bounded (or compact, or weakly compact) operators from certain Banach spaces into $C\left(X\right)$.

Standard facts about separating linear functionals will be used to determine how two cones $C$ and $D$ and their duals $C^{*}$ and $D^{*}$ may overlap. When $TV\to W$ is linear and $K\subset V$ and $D\subset W$ are cones, these results will be applied to $C=T\left(K\right)$ and $D$, giving a unified treatment of several theorems of the alternate which explain when $C$ contains an interior point of $D$. The case when $V=W$ is the space $H$ of $n\times n$ Hermitian matrices, $D$ is the $n\times n$ positive semidefinite matrices, and $T\left(X\right)=AX+X^{*}A$ yields new and known results about the existence of block diagonal...

We examine the so-called three-space-stability for some classes of linear topological and locally convex spaces for which this problem has not been investigated.

The aim of this paper is an investigation of topological algebras with an orthogonal sequence which is total. Closed prime ideals or closed maximal ideals are kernels of multiplicative functionals and the continuous multiplicative functionals are given by the “coefficient functionals”. Our main result states that an orthogonal total sequence in a unital Fréchet algebra is already a Schauder basis. Further we consider algebras with a total sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ satisfying $x_n^2=x_n$ and $x_n{x}_{n+1}=x_{n+1}$ for all n ∈ ℕ.

It is shown that all maximal regular ideals in a Hausdorff topological algebra A are closed if the von Neumann bornology of A has a pseudo-basis which consists of idempotent and completant absolutely pseudoconvex sets. Moreover, all ideals in a unital commutative sequentially Mackey complete Hausdorff topological algebra A with jointly continuous multiplication and bounded elements are closed if the von Neumann bornology of A is idempotently pseudoconvex.