Uniform Boundedness Principle Research Articles

Versions of Eberlein-Smulian and Amir-Lindenstrauss theorems in the framework of conditional sets

Based on conditional set theory, we study conditional weak topologies, extending some well-known results to this framework and culminating with the proof of conditional versions of Eberlein-Šmulian and Amir-Lindenstrauss theorems. In pursuing this aim, we prove conditional versions of Baire Category theorem and Uniform Boundedness Principle.

Logical metatheorems for abstract spaces axiomatized in positive bounded logic

In this paper we show that normed structures which can be axiomatized in positive bounded logic (in the sense of Henson and Iovino) admit proof-theoretic metatheorems (as developed by the second author since 2005) on the extractability of explicit uniform bounds from proofs in the respective theories. We apply this to design such metatheorems for abstract Banach lattices, Lp- and C(K)-spaces as well as bands in Lp(Lq)-Bochner spaces. We also show that a proof-theoretic uniform boundedness principle can serve in many ways as a substitute for the model-theoretic use of ultrapowers of Banach spaces.

A simple proof of the closed graph theorem

<p>Assume that <em>A</em> is a closed linear operator defined on all of a Hilbert space <em>H</em>. Then, <em>A</em> is bounded. This classical theorem is proved on the basis of uniform boundedness principle. The proof is easily extended to Banach spaces.</p>

The Banach-Steinhaus Theorem in Abstract Duality Pairs

Let E, F be sets and G a Hausdorff, abelian topological group with b : E X F→ G; we refer to E, F, G as an abstract duality pair with respect to G or an abstract triple and denote this by (E,F : G). Let (Ei,Fi : G) be abstract triples for i = 1, 2. Let Fi be a family of subsets of Fi and let τFi(Ei) = τibe the topology on Ei of uniform convergence on the members of Fi. Let J be a family of mappings from Ei to E2. We consider conditions which guarantee that J is τ1-τ2equicontinuous. We then apply the results to obtain versions of the Banach-Steinhaus Theorem for both abstract triples and for linear operators between locally convex spaces.

Fuzzy Continuous Mappings in Fuzzy Normed Linear Spaces

In this paper we continue the study of fuzzy continuous mappings in fuzzy normed linear spaces initiated by T. Bag and S.K. Samanta, as well as by I. Sadeqi and F.S. Kia, in a more general settings. Firstly, we introduce the notion of uniformly fuzzy continuous mapping and we establish the uniform continuity theorem in fuzzy settings. Furthermore, the concept of fuzzy Lipschitzian mapping is introduced and a fuzzy version for Banach’s contraction principle is obtained. Finally, a special attention is given to various characterizations of fuzzy continuous linear operators. Based on our results, classical principles of functional analysis (such as the uniform boundedness principle, the open mapping theorem and the closed graph theorem) can be extended in a more general fuzzy context.

The uniform boundedness principles for γ -max-pseudo-norm-subadditive and quasi-homogeneous operators in F∗ spaces

The uniform boundedness principles for γ -max-pseudo-norm-subadditive and quasi-homogeneous operators in F∗ spaces

Strong divergence for system approximations

In this paper we analyze approximation of stable linear time-invariant systems, like the Hilbert transform, by sampling series for bandlimited functions in the Paley–Wiener space PW π 1 . It is known that there exist systems and functions such that the approximation process is weakly divergent, i.e., divergent for certain subsequences. Here we strengthen this result by proving strong divergence, i.e., divergence for all subsequences. Further, in case of divergence, we give the divergence speed. We consider sampling at Nyquist rate as well as oversampling with adaptive choice of the kernel. Finally, connections between strong divergence and the Banach–Steinhaus theorem, which is not powerful enough to prove strong divergence, are discussed.

BOUNDEDNESS OF LINEAR ORDER-HOMOMORPHISMS IN L-TOPOLOGICAL VECTOR SPACES

A new definition of boundedness of linear order-homomorphisms (LOH)in $L$-topological vector spaces is proposed. The new definition iscompared with the previous one given by Fang [The continuity offuzzy linear order-homomorphism, J. Fuzzy Math. 5 (4) (1997)829$-$838]. In addition, the relationship between boundedness andcontinuity of LOHs is discussed. Finally, a new uniform boundednessprinciple in $L$-topological vector spaces is established in thesense of a new definition of uniform boundedness for a family ofLOHs.

On the uniform boundedness theorem in fuzzy quasi-normed spaces

We prove that a family of continuous linear operators from a fuzzy quasi-normed space of the half second category to a fuzzy quasi-normed space is uniformly fuzzy bounded if and only if it is pointwise fuzzy bounded. This result generalizes and unifies several well-known results; in fact, the classical uniform boundedness principle, or Banach–Steinhauss theorem, is deduced as a particular case. Furthermore, we establish the relationship between uniform fuzzy boundedness and equicontinuity which allows us to give a uniform boundedness theorem in the class of paratopological vector spaces. The classical result for topological vector spaces is deduced as a corollary.

Zabrejko's lemma and the fundamental principles of functional analysis in the asymmetric case

Some versions of the fundamental principles of the functional analysis in asymmetric normed spaces – the Open Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Principle – are proved. The proofs are based on an asymmetric version of a lemma of Zabreijko (1969) [12] on the continuity of the countably subadditive functionals. At the same time a flaw in the proof of the Uniform Boundedness Principle, given in the book by Cobzaş (2013) [5], is fixed.

Bidual Spaces and Reflexivity of Real Normed Spaces

Summary In this article, we considered bidual spaces and reflexivity of real normed spaces. At first we proved some corollaries applying Hahn-Banach theorem and showed related theorems. In the second section, we proved the norm of dual spaces and defined the natural mapping, from real normed spaces to bidual spaces. We also proved some properties of this mapping. Next, we defined real normed space of R, real number spaces as real normed spaces and proved related theorems. We can regard linear functionals as linear operators by this definition. Accordingly we proved Uniform Boundedness Theorem for linear functionals using the theorem (5) from [21]. Finally, we defined reflexivity of real normed spaces and proved some theorems about isomorphism of linear operators. Using them, we proved some properties about reflexivity. These formalizations are based on [19], [20], [8] and [1].

Subseries convergence in abstract duality pairs

Let E, F be sets, G an Abelian topological group and b : ExF — G. Then (E, F, G) is called an abstract triple. Let w(F, E) be the weakest toplogy on F such that the maps {b(x, ·): x G E} from F into G are continuous. A subset B C F is w(F,E) sequentially conditionally compact if every sequence {yk} C B has a subsequence {ynk } such that limj; b(x, ynk) exists for every x G E. It is shown that if a formal series in E is subseries convergent in the sense that for every subsequence {xnj} there is an element x G E such that Xj=! b(xnj ,y) = b(x,y) for every y G F ,then the series Xj=! b(xnj ,y) converge uniformly for y belonging to w(F, E) sequentially conditionally compact subsets ofF. This result is used to establish Orlicz-Pettis Theorems in locall convex and function spaces. Applications are also given to Uniform Boundedness Principles and continuity results for bilinear mappings.

SOME PROPERTIES OF FUZZY NORM OF LINEAR OPERATORS

In the present paper, we study some properties of fuzzy norm of linear operators. At rst the bounded inverse theorem on fuzzy normed linear spaces is investigated. Then, we prove Hahn Banach theorem, uniform boundedness theorem and closed graph theorem on fuzzy normed linear spaces. Finally the set of all compact operators on these spaces is studied.

Strong divergence of reconstruction procedures for the Paley–Wiener space [formula omitted] and the Hardy space [formula omitted

Previous results on certain sampling series have left open if divergence only occurs for certain subsequences or, in fact, in the limit. Here we prove that divergence occurs in the limit.We consider three canonical reconstruction methods for functions in the Paley–Wiener space PWπ1. For each of these we prove an instance when the reconstruction diverges in the limit. This is a much stronger statement than previous results that provide only lim sup divergence. We also address reconstruction for functions in the Hardy space H1 and show that for any subsequence of the natural numbers there exists a function in H1 for which reconstruction diverges in lim sup. For two of these sampling series we show that when divergence occurs, the sampling series has strong oscillations so that the maximum and the minimum tend to positive and negative infinity. Our results are of interest in functional analysis because they go beyond the type of result that can be obtained using the Banach–Steinhaus Theorem. We discuss practical implications of this work; in particular the work shows that methods using specially chosen subsequences of reconstructions cannot yield convergence for the Paley–Wiener Space PWπ1.

Transversal functional analysis

This paper provides an introduction to the ideas and methods of transversal functional analysis based on the transversal sets theory. A unifying concept that lies at the heart of transversal functional analysis is that of a transversal normed linear space. I have developed the theory far enough to include facts of have called the three new basic principles of linear analysis as: Form of Hahn-Banach theorem, Form of Principle of Uniform Boundedness (= Form of Banach-Steinhaus theorem), and Form of Open Mapping theorem. In the classical functional analysis fundamental fact is Riesz lemma. In transversal functional analysis (on lower transversal normed spaces) its role play so-called Geometrical lemma! This paper presents applications of the Axiom of Infinite Choice.

On the sharpness of an error bound for a Galerkin method to solve parabolic differential equations

This paper discusses the sharpness of an error bound for the standard Galerkin method for the approximate solution of a parabolic differential equation. A backward difference is used for discretization in time, and a variational method like the finite element method is considered for discretization in space. The error bound is written in terms of an averaged modulus of continuity. Whereas the direct estimate follows by standard methods, the sharpness of the bound is established by an application of a quantitative extension of the uniform boundedness principle as proposed in Dickmeis et al. (1984) [4].

A Sharp Error Estimate for Numerical Fourier Fransform of Band-Limited Functions Based on Windowed Samples

W. Dickmeis and R. J. Nessel published their first version of a quantitative extension of the classical uniform boundedness principle in [J. Approx. Theory 31 (1981), 161–174]. It is a general approach to finding counterexamples that prove sharpness of error estimates. So far applications of this principle include error bounds for approximation processes, cubature rules, ordinary and partial differential equations, and reconstruction from samples. Here we discuss the error of discrete approximations of the Fourier transform based on windowed samples for band-limited functions. The results can be applied to the Hann- and Blackmann–Harris-window but also to window-functions that enable higher orders of convergence. We describe a class of such windows.

The Minimal Context for Local Boundedness in Topological Vector Spaces

Abstract The local boundedness of classes of operators is analyzed on different subsets directly related to the Fitzpatrick function associated to an operator. Characterizations of the topological vector spaces for which that local boundedness holds is given in terms of the uniform boundedness principle. For example the local boundedness of a maximal monotone operator on the algebraic interior of its domain convex hull is a characteristic of barreled locally convex spaces.

Topological Methods in Analysis

Topological methods have played a seminal role in functional analysis since its birth in the early twentieth century. The Baire category theorem, for example, is the bedrock on which rest such basic principles of functional analysis as the open mapping theorem and the principle of uniform boundedness. Initial (weak) topologies, compactness, and the Tikhonov theorem drive such classical results of duality theory as the Banach-Alaoglu and Krein-Milman theorems. Topological methods also play a crucial role in Banach algebra theory (Gelfand topology), harmonic analysis (locally compact groups and function spaces), differential equations (fixed point theorems and Ascoli-Arzela theorem), and nonlinear analysis (fixed point existence theorems and topological degree theory) to mention just a few. The lofty goal of this special issue is to present some new outstanding contributions in this research area. They include results on fixed point theorems, metric spaces, C-algebras, modular spaces, algebraic hypersurfaces, and topological groups. Our hope is that this wide spectrum of topics will be of interest to many researchers in the area.

On the Banach-Steinhaus theorem

A Banach-Steinhaus theorem for sets of continuous linear mappings on topological modules which are Baire spaces is proved, and some consequences of it are derived.