Properties of the so called ${\theta }_o$-complete topological spaces are investigated. Also, necessary and sufficient conditions are given so that the space $C(X,E)$ of all continuous functions, from a zero-dimensional topological space $X$ to a non-Archimedean locally convex space $E$, equipped with the topology of uniform convergence on the compact subsets of $X$ to be polarly barrelled or polarly quasi-barrelled.

Necessary and sufficient conditions are given so that the space $C(X,E)$ of all continuous functions from a zero-dimensional topological space $X$ to a non-Archimedean locally convex space $E$, equipped with the topology of uniform convergence on the compact subsets of $X$, to be polarly absolutely quasi-barrelled, polarly ${\aleph }_o$-barrelled, polarly ${\ell }^{\infty }$-barrelled or polarly $c_o$-barrelled. Also, tensor products of spaces of continuous functions as well as tensor products of certain $E^{\prime }$-valued measures are investigated.

We prove that for each linear contraction T : X → X (∥T∥ ≤ 1), the subspace F = {x ∈ X : Tx = x} of fixed points is 1-complemented, where X is a suitable subspace of L¹(E*) and E* is a separable dual space such that the weak and weak* topologies coincide on the unit sphere. We also prove some related fixed point results.

In this paper we bring together the different known ways of establishing the continuity of the integral over a uniformly integrable set of functions endowed with the topology of pointwise convergence. We use these techniques to study Pettis integrability, as well as compactness in C(K) spaces endowed with the topology of pointwise convergence on a dense subset D in K.

Suppose $E$ is an ordered locally convex space, $X_1$ and $X_2$ Hausdorff completely regular spaces and $Q$ a uniformly bounded, convex and closed subset of $M_t^{+}(X_1\times X_2,E)$. For $i=1,2$, let ${\mu }_i\in M_t^{+}(X_i,E)$. Then, under some topological and order conditions on $E$, necessary and sufficient conditions are established for the existence of an element in $Q$, having marginals ${\mu }_1$ and ${\mu }_2$.

For $i=(1,2)$, let $X_i$ be completely regular Hausdorff spaces, $E_i$ quasi-complete locally convex spaces, $E=E_1\breve{\otimes }{E}_2$, the completion of the their injective tensor product, $C_b\left(X_i\right)$ the spaces of all bounded, scalar-valued continuous functions on $X_i$, and ${\mu }_i$$E_i$-valued Baire measures on $X_i$. Under certain...

An Orlicz-Pettis type property for vector measures and also the “Uniform Boundedness Principle” are shown to fail without local convexity assumption. The author asks under which generalized convexity hypotheses these properties remain true. This problem is expressed in terms of barrelledness type conditions.