Bornological spaces of non-Archimedean valued functions

Let C(X,E) be the space of all continuous functions from an ultraregular space X to a non-Archimedean locally convex space E. Necessary and/or sufficient conditions are given so that C(X,E), with the topology of uniform convergence on compact sets or with the topology of simple convergence, is bornological or c-ultrabornological.     

Ordered representations of spaces of integrable functions

Let X be a Banach space, (Ω,Σ) a measurable space and let m : Σ → X be a (countably additive) vector measure. Consider the corresponding space of integrable functions L¹(m). In this paper we analyze the set of (countably additive) vector measures n satisfying that L¹(n) = L¹(m). In order to do this we define a (quasi) order relation on this set to obtain under adequate requirements the simplest representation of the space L¹(m) associated to downward directed subsets of the set of all the representations. This research has been partially supported by La Junta de Andalucía. The support of D.G.I. under project MTM2006–11690–C02 (M.E.C. Spain) and FEDER is gratefully acknowledged.     

Approximations in spaces of locally integrable functions

We study approximations of functions from the sets $\hat L_\beta ^\psi \mathfrak{N}$ , which are determined by convolutions of the following form: $$f\left( x \right) = A_0 + \int\limits_{ - \infty }^\infty {\varphi \left( {x + t} \right)\hat \psi _\beta \left( t \right)dt, \varphi \in \mathfrak{N}, \hat \psi _\beta \in L\left( { - \infty ,\infty } \right),} $$ where η is a fixed subset of functions with locally integrablepth powers (p≥1). As approximating aggregates, we use the so-called Fourier operators, which are entire functions of exponential type ≤ σ. These functions turn into trigonometric polynomials if the function ?(·) is periodic (in particular, they may be the Fourier sums of the function approximated). The approximations are studied in the spacesL _p determined by local integral norms ∥·∥_-p . Analogs of the Lebesgue and Favard inequalities, wellknown in the periodic case, are obtained and used for finding estimates of the corresponding best approximations which are exact in order. On the basis of these inequalities, we also establish estimates of approximations by Fourier operators, which are exact in order and, in some important cases, exact with respect to the constants of the principal terms of these estimates.     

Bornological properties of spaces of non-Archimedean valued functions

Summary Let C(X, K) be the space of continuous functions from an ultraregular space X into a non-Archimedean field K. C(X, K) is a locally convex space with semi-norms ·_F where F belongs to a familyT of bounding subsets of theZ-repletion v₀X of X. We give necessary and sufficient conditions for C(X, K) to be bornological, semi-bornological or ultrabornological.

---

Résumé Soit C(X, K) l'espace des fonctions continues sur l'espace ultraregulière X avec valeurs dans le corps non-Archimédien K. C(X, K) est un espace localement convexe avec semi-normes ·_F où F appartient à une familleT de sous-ensembles bornantes de laZ-replété_v0X de X. Nous donnons des conditions nécessaires et suffisantes pour que X soit bornologigue, semi-bornologique ou ultrabornologique.

---

---

«Aangesteld navorser» of the Belgian «Nationaal Fonds voor Wetenschappelijk Onderzoek».     

Isometries on spaces of vector valued Lipschitz functions

This paper gives a characterization of a class of surjective isometries on spaces of Lipschitz functions with values in a finite dimensional complex Hilbert space.     

Distances to spaces of measurable and integrable functions

Given a complete probability space and a Banach space X we establish formulas to compute the distance from a function to the spaces of strongly measurable functions and Bochner integrable functions. We study the relationship between these distances and use them to prove some quantitative counterparts of Pettis’ measurability theorem. We also give several examples showing that some of our estimates are sharp.     

On the ideal centre of the space of vector valued integrable functions

Let E be a Banach lattice and L¹(μ, E) be the space of E-valued Bochner integrable functions. Some order properties of L¹(μ, E) are given. It is shown that L_s^∞(μ, Z(E)) is the ideal centre of L¹(μ, E) and it is obtained a Radon-Nikodym type theorem for B -integrable functions.      

Weak continuity of Riemann integrable functions in Lebesgue-Bochner spaces

In general, Banach space-valued Riemann integrable functions defined on [0, 1] （equipped with the Lebesgue measure） need not be weakly continuous almost everywhere. A Banach space is said to have the weak Lebesgue property if every Riemann integrable function taking values in it is weakly continuous almost everywhere. In this paper we discuss this property for the Banach space LX^1 of all Bochner integrable functions from [0, 1] to the Banach space X. We show that LX^1 has the weak Lebesgue property whenever X has the Radon-Nikodym property and X＊ is separable. This generalizes the result by Chonghu Wang and Kang Wan [Rocky Mountain J. Math., 31（2）, 697-703 （2001）] that L^1[0, 1] has the weak Lebesgue property.     

Inductive and projective limits of Banach spaces of measurable functions with order unities with respect to power parameter

We prove that a measurable function f is bounded and invertible if and only if there exist at least two equivalent norms by order unit spaces with order unities f^α and f^β with α > β > 0. We show that it is natural to understand the limit of ordered vector spaces with order unities f^α (α approaches to infinity) as a direct sum of one inductive and one projective limits. We also obtain some properties for the corresponding limit topologies.     

On the convergence of series in spaces of integrable functions

A sufficient condition for the convergence of series in the spaces L _p on a set of infinite measure is obtained.     

Extensions of certain operators to spaces of abstract integrable functions

In this paper we discuss the extension of operators onL ₁ ^R spaces to operators onL ₁ ^E andP ₁ ^E spaces (see Section 1), whereE is a Banach space. A necessary and sufficient condition for the existence of the extension to a spaceP ₁ ^E is given (see Section 3) whenE has the weak Radon-Nikodym property. The paper contains certain applications to ergodic theory and a theorem giving a characterization of weakly conditionally compact sets.     

Radial limits of inner functions and Bloch spaces

Let be an inner function in the unit ball , . Assume that

where and is the radial derivative. Then, for all , the set has a non-zero real Hausdorff -content, and it has a non-zero complex Hausdorff -content.