A counterexample on non-archimedean regularity

A non-regular inductive sequence of non-archimedean reflexive Fréchet spaces is constructed. On the other hand, it is proved that every inductive sequence of reflexive Banach spaces over a spherically complete field is regular. Also, some applications are given.     

Duality and Reflexivity of Spaces of Approximable Polynomials on Locally Convex Spaces

 We develop a duality theory for spaces of approximable n-homogeneous polynomials on locally convex spaces, generalising results previously obtained for Banach spaces. For E a Fréchet space with its dual having the approximation property and with E′ _b locally Asplund we show that the space of n-homogeneous polynomials on (E′ _b )′ _b is the inductive dual of the space of boundedly weakly continuous n-homogeneous polynomials on E. We show that when E is a reflexive Fréchet space, the space of n-homogeneous polynomials on E is reflexive if and only if every n-homogeneous polynomial on E is boundedly weakly continuous. (Received 24 March 1999; in final form 14 February 2000)     

Reflexivity of Inductive Limits

An inductive locally convex limit of reflexive topological spaces is reflexive iff it is almost regular.     

Characterization of reflexivity by equivalent renorming

A new rotundity property of Day's norm on c₀(Γ) is introduced. This property provides in particular a renorming characterization of the class of all reflexive Banach spaces.     

Mackey Topologies and Mixed Topologies in Riesz Spaces

The possibility of characterizing the Mackey topology of a dual pair of vector spaces as a generalized inductive limit (or mixed) topology is investigated. Positive answers are given for a wide range of dual pairs of Riesz spaces (vector lattices) and non-commutative Banach function spaces (or symmetric operator spaces).     

ConvergenceLF spaces

In this paper we examineLF spaces, inductive limits of Fréchet spaces, in two different settings: the categoryCV S of convergence vector spaces and the categoryLC S of locally convex topological vector spaces. Special attention is given to permanence properties and retractivity properties in each case. Some interaction between properties ofLF spaces inCV S and other properties inLC S are investigated.R. Beattie's research was supported by NSERC grant OGP0005316.     

Reflexive operators with domain in Köthe spaces

It is proved that if a K?the space λ₁(A) is distinguished and E is an arbitrary Fréchet space then every reflexive map T: λ₁(A)→E (i.e., T maps bounded sets into relatively weakly compact ones) factorizes through a reflexive Fréchet space. An analogous result is proved for Montel maps (i.e., which map bounded sets into relatively compact ones). The result is a consequence of the fact proved also in this paper that, for a distinguished λ₁(A) space, the spaces of reflexive maps R(λ₁(A), C(K)) and of Montel maps M(λ₁(A), C(K)) are the Mackey completions of the spaces of weakly compact and compact maps, respectively. Consequences for spaces of vector-valued (weakly) continuous functions are also obtained. Received: 24 November 1997 / Revised version: 14 May 1998     

Some new rich subspaces of C. Applications

In this paper, we are interested in a class of subspaces of C, introduced by Bourgain [Studia Math. 77 (1984) 245-253]. Wojtaszczyk called them rich in his monograph [Banach Spaces for Analysts, Cambridge Univ. Press, 1991]. We give some new examples of such spaces: this allows us to recover previous results of Godefroy-Saab and Kysliakov on spaces with reflexive annihilator in a very simple way. We construct some other examples of rich spaces, hence having property (V) of Pe?czyński and Dunford-Pettis property. We also recover the results due to Bourgain and Saccone saying that spaces of uniformly convergent Fourier series share these properties, by only using the main result of [Studia Math. 77 (1984) 245-253] and some very elementary arguments. We generalize too these results.     

Spaces of p-integrable Functions with Respect to a Vector Measure

We study several properties of the Banach lattices L^p (m) and L^p_w (m) of p-integrable scalar functions and weakly p-integrable scalar functions with respect to a countably additive vector measure m. The relation between these two spaces plays a fundamental role in our analysis. This research has been partially supported by La Consejería de Educatión y Ciencia de la Junta de Andalucía.     

Some properties of maximal monotone operators on nonreflexive Banach spaces

Important properties of maximal monotone operators on reflexive Banach spaces remain open questions in the nonreflexive case. The aim of this paper is to investigate some of these questions for the proper subclass of locally maximal monotone operators. (This coincides with the class of maximal monotone operators in reflexive spaces.) Some relationships are established with the maximal monotone operators of dense type, which were introduced by J.-P. Gossez for the same purpose.     

De Branges Spaces of Entire Functions Symmetric About the Origin

We define and investigate the class of symmetric and the class of semibounded de Branges spaces of entire functions. A construction is made which assigns to each symmetric de Branges space a semibounded one. By employing operator theoretic tools it is shown that every semibounded de Branges space can be obtained in this way, and which symmetric spaces give rise to the same semibounded space. Those subclasses of Hermite-Biehler functions are determined which correspond to symmetric or semibounded, respectively, nondegenerated de Branges spaces. The above assignment is determined in terms of the respective generating Hermite-Biehler functions.     

Local rotundity structure of generalized Orlicz–Lorentz sequence spaces

We give some criteria for extreme points and strong U-points in generalized Orlicz–Lorentz sequence spaces, which were introduced in [P. Foralewski, H. Hudzik, L. Szymaszkiewicz, On some geometric and topological properties of generalized Orlicz–Lorentz sequence spaces, Math. Nachr. (in press)] (cf. [G.G. Lorentz, An inequality for rearrangements, Amer. Math. Monthly 60 (1953) 176–179; M. Nawrocki, The Mackey topology of some F-spaces, Ph.D. Dissertation, Adam Mickiewicz University, Poznań, 1984 (in Polish)]). Some examples show that in these spaces the notion of the strong U-point is essentially stronger than the notion of the extreme point. This paper is related to the results from [A. Kamińska, Extreme points in Orlicz–Lorentz spaces, Arch. Math. 55 (1990) 173–180] (see Remark 1).     

Countable inductive limits of frechet algebras

We show that the Gelfand-Mazur theorem holds for countable inductive limits of Frechet algebras (we do not assume that the homomorphisms which define the inductive limit are continuous, or one-to-one). This question is motivated by the fact that the spectrum of some elements of such an algebra may be empty. We also discuss in detail a countable inductive limit of Frechet algebras of holomorphic functions, which provides an elementary, but seminal, counterexample to the biinvariant subspace problem for complete, reflexive, locally convex spaces.     

Genericity and amalgamation of classes of Banach spaces

We study universality problems in Banach space theory. We show that if A is an analytic class, in the Effros-Borel structure of subspaces of C([0,1]), of non-universal separable Banach spaces, then there exists a non-universal separable Banach space Y, with a Schauder basis, that contains isomorphs of each member of A with the bounded approximation property. The proof is based on the amalgamation technique of a class C of separable Banach spaces, introduced in the paper. We show, among others, that there exists a separable Banach space R not containing L¹(0,1) such that the indices β and rND are unbounded on the set of Baire-1 elements of the ball of the double dual R∗∗ of R. This answers two questions of H.P. Rosenthal.We also introduce the concept of a strongly bounded class of separable Banach spaces. A class C of separable Banach spaces is strongly bounded if for every analytic subset A of C there exists Y∈C that contains all members of A up to isomorphism. We show that several natural classes of separable Banach spaces are strongly bounded, among them the class of non-universal spaces with a Schauder basis, the class of reflexive spaces with a Schauder basis, the class of spaces with a shrinking Schauder basis and the class of spaces with Schauder basis not containing a minimal Banach space X.     

Commutativity of the Valdivia–Vogt table of representations of function spaces

In this article, we show that the Valdivia–Vogt structure table—containing the sequence space representations of the most used spaces of smooth functions appearing in the theory of distributions—can be interpreted as a commutative diagram, i.e., there is an isomorphism between the space of infinitely differentiable functions and the space , where s is the space of rapidly decreasing sequences, such that its restriction to the other function spaces in the structure table yields an isomorphism between these spaces of smooth functions and their sequence space representation. This result answers the corresponding question of Prof. Dietmar Vogt formulated on the conference “Functional Analysis: Applications to Complex Analysis and Partial Differential Equations” held in B?dlewo in May 2012.     

Interpolation of Bilinear Operators Between Quasi-Banach Spaces

We study interpolation, generated by an abstract method of means, of bilinear operators between quasi-Banach spaces. It is shown that under suitable conditions on the type of these spaces and the boundedness of the classical convolution operator between the corresponding quasi-Banach sequence spaces, bilinear interpolation is possible. Applications to the classical real method spaces, Calderón-Lozanovsky spaces, and Lorentz-Zygmund spaces are presented. The author is supported by the National Science Foundation under grant DMS 0099881. The author is supported by KBN Grant 1 P03A 013 26.     

Projective limits of weighted (LB)-spaces of continuous functions

Countable projective spectra of countable inductive limits, called (PLB)-spaces, of weighted Banach spaces of continuous functions are investigated. It is characterized when the derived projective limit functor vanishes in terms of the sequences of the weights defining the spaces. The locally convex properties of the corresponding projective limits are analyzed, too. Received: 30 January 2009     

The gliding hump property in vector sequence spaces

It is shown that vector sequence spaces with a gliding hump property have many of the properties of complete spaces. For example, it is shown that the -dual of certain vector sequence spaces with a gliding hump property are sequentially complete with respect to the topology of pointwise convergence and also versions of the Banach-Steinhaus Theorem are established for such spaces.     

The Goldstine Theorem for asymmetric normed linear spaces

It is well known that if (X,q) is an asymmetric normed linear space, then the function qs defined on X by qs(x)=max{q(x),q(−x)}, is a norm on the linear space X. However, the lack of symmetry in the definition of the asymmetric norm q yields an algebraic asymmetry in the dual space of (X,q). This fact establishes a significant difference with the standard results on duality that hold in the case of locally convex spaces. In this paper we study some aspects of a reflexivity theory in the setting of asymmetric normed linear spaces. In particular, we obtain a version of the Goldstine Theorem to these spaces which is applied to prove, among other results, a characterization of reflexive asymmetric normed linear spaces.