Properties of the so called -complete topological spaces are investigated. Also, necessary and sufficient conditions are given so that the space of all continuous functions, from a zero-dimensional topological space to a non-Archimedean locally convex space , equipped with the topology of uniform convergence on the compact subsets of to be polarly barrelled or polarly quasi-barrelled.
Necessary and sufficient conditions are given so that the space of all continuous functions from a zero-dimensional topological space to a non-Archimedean locally convex space , equipped with the topology of uniform convergence on the compact subsets of , to be polarly absolutely quasi-barrelled, polarly -barrelled, polarly -barrelled or polarly -barrelled. Also, tensor products of spaces of continuous functions as well as tensor products of certain -valued measures are investigated.
We prove that if E is a subset of a Banach space whose density is of measure zero and such that (E, weak) is a paracompact space, then (E, weak) is a Radon space of type (F ) under very general conditions.
We present several characterizations and representations of semi-complete vector fields on the open unit balls in complex Euclidean and Hilbert spaces.
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.
We consider the multiplicative algebra P(𝒢₊') of continuous scalar polynomials on the space 𝒢₊' of Roumieu ultradistributions on [0,∞) as well as its strong dual P'(𝒢₊'). The algebra P(𝒢₊') is densely embedded into P'(𝒢₊') and the operation of multiplication possesses a unique extension to P'(𝒢₊'), that is, P'(𝒢₊') is also an algebra. The operation of differentiation on these algebras is investigated. The polynomially extended Laplace transformation and its connections with the differentiation...