Uniform Boundedness Principle Research Articles

Extended locally convex spaces: Barreledness, equicontinuity and Banach-Steinhaus theorem

For an extended locally convex space (X,τ), in [12], the authors studied the finest locally convex topology (flc topology) τF on X coarser than τ. One can often prove facts about (X,τ) by applying classical locally convex space theory on (X,τF). This paper employs the flc topology to analyze barreled extended locally convex spaces and establish a version of the Banach-Steinhaus theorem in the extended setting. One of the key results of this paper is the relationship between the barreledness of an extended locally convex space (X,τ) and the barreledness of the associated finest locally convex space (X,τF). This is achieved by examining the lower semi-continuous seminorms on these spaces.

Order boundedness and order continuity properties of positive operator semigroups

Relatively uniformly continuous (ruc) semigroups were recently introduced and studied by Kandic, Kramar-Fijavž, and the second-named author, in order to make the theory of one-parameter operator semigroups available in the setting of vector lattices, where no norm is present in general. In this article, we return to the more standard Banach lattice setting – where both ruc semigroups and C 0 -semigroups are well-defined concepts – and compare both notions. We show that the ruc semigroups are precisely those positive C 0 -semigroups whose orbits are order bounded for small times. We then relate this result to three different topics: (i) equality of the spectral and the growth bound for positive C 0 -semigroups; (ii) a uniform order boundedness principle which holds for all operator families between Banach lattices; and (iii) a description of unbounded order convergence in terms of almost everywhere convergence for nets which have an uncountable index set containing a co-final sequence.

On the Extension of Bounded Holomorphic Maps from Gleason Parts of the Maximal Ideal Space of

Abstract Let $H^\infty $ be the algebra of bounded holomorphic functions on the open unit disk, and let $\mathfrak M$ be its maximal ideal space. Let $\mathfrak M_a$ be the union of nontrivial Gleason parts (analytic disks) of $\mathfrak M$ . In this paper, we study the problem of extensions of bounded Banach-valued holomorphic functions and holomorphic maps with values in Oka manifolds from Gleason parts of $\mathfrak M_a\setminus \mathbb {D}$ . The resulting extensions satisfy the uniform boundedness principle in the sense that their norms are bounded by constants that do not depend on the choice of the Gleason part. The results extend fundamental results of D. Suárez on the characterization of the algebra of restrictions of Gelfand transforms of functions in $H^\infty $ to Gleason parts of $ \mathfrak M_a\setminus \mathbb {D}$ . The proofs utilize our recent advances on $\bar \partial $ -equations on quasi-interpolating sets and Runge-type approximations.

Applications of uniform boundedness principle to matrix transformations

Using the uniform boundedness principle of Maddox, we characterize matrix transformations from the space $(\ell_{p}) _{T}$ to the spaces $m(\phi )$ and $n(\phi )$ for the case $1\leq p\leq \infty$, which correspond to bounded linear operators. Here $(\ell _{p})_{T}$ is the domain of an arbitrary triangle matrix $T$ in the space $\ell _{p}$, and the spaces $m(\phi )$ and $n(\phi )$ are introduced by W.L.C. Sargent. In special cases, we get some well known results of W.L.C. Sargent, M. Stieglitz and H. Tietz, E. Malkowsky and E. Savaş. Also we give other applications including some important new classes.

Open mapping theorem for nonlinear operators

Many authors gave definitions of a new kind of derivative and integral, moving the roles of subtraction and addition to division and multiplication, and thus established a new calculus, called multiplicative calculus. In this paper, our aim is to bring up this calculus to the attention of researchers and demonstrate its usefulness. We first discussed multiplicative normed spaces by giving some topological properties of the relevant multiplicative normed spaces. Also, we give a generalization of the open mapping theorem and the uniform boundedness principle for nonlinear operators with a number of important examples.

On convergence of Fourier series in discrete Jacobi–Sobolev spaces

In this paper, we show a complete characterization of the uniform boundedness of the partial sum operator in a discrete Sobolev space with Jacobi measure. As a consequence, we obtain the convergence of the Fourier series. Moreover it is showed that this Sobolev space is the first category which implies that it is not possible to apply the Banach–Steinhaus theorem.

New families satisfying the dynamical uniform boundedness principle over function fields

We extend a technique, originally due to the first author and Poonen, for proving cases of the Strong Uniform Boundedness Principle (SUBP) in algebraic dynamics over function fields of positive characteristic. The original method applied to unicritical polynomials for which the characteristic does not divide the degree. We show that many new 1-parameter families of polynomials satisfy the SUBP, including the family of all quadratic polynomials in even characteristic. We also give a new family of non-polynomial, non-Lattès rational functions that satisfies the SUBP.

From Uniform Boundedness to the Boundary Between Convergence and Divergence

Summary We introduce a dual of the uniform boundedness principle that does not require completeness and gives an indirect means for testing the boundedness of a set. The dual principle, although known to analysts and despite its applications in establishing results such as the Hellinger-Toeplitz theorem, is often missing from elementary treatments of functional analysis. We give an example showing a connection between the dual principle and a question in the spirit of du Bois-Reymond regarding the boundary between convergence and divergence for sequences. This example is intended to illustrate why the statement of the principle is natural and clarify what the principle claims and what it does not.

Fuzzy Continuous Mappings on Fuzzy F-Spaces

In the present paper, we first introduce different types of fuzzy continuity for mappings between fuzzy F-normed linear spaces and the relations between them are investigated. Secondly, the principles of fuzzy functional analysis are established in the context of fuzzy F-spaces. More precisely, based on the fact that fuzzy continuity and topological continuity are equivalent, we obtain the closed graph theorem and the open mapping theorem. Using Zabreiko’s lemma, we prove the uniform bounded principle and Banach–Steinhaus theorem. Finally, some future research directions are presented.

On Special Properties for Continuous Convex Operators and Related Linear Operators

This paper provides a uniform boundedness theorem for a class of convex operators, such as Banach–Steinhaus theorem for families of continuous linear operators. The case of continuous symmetric sublinear operators is outlined. Second, a general theorem characterizing the existence of the solution of the Markov moment problem is reviewed, and a related minimization problem is solved. Convexity is the common point of the two aims of the paper mentioned above.

Banach-Steinhaus theorem for linear relations on asymmetric normed spaces

We study the continuity of linear relations defined on asymmetric normed spaces with values in normed spaces. We give some geometric charactirization of these mappings. As an application, we prove the Banach-Steinhaus theorem in the framework of asymmetric normed spaces.

Representation of uniform boundedness principle and Hahn–Banach theorem in linear n-normed space

The concept of b-linear functional and its different types of continuity in linear n-normed space are presented and some of their properties are being established. We derive the Uniform Boundedness Principle and Hahn-Banach extension Theorem with the help of bounded b-linear functionals in the case of linear n-normed spaces and discuss some examples and applications. Finally, we present the concept of weak*convergence for the sequence of bounded b-linear functionals in linear n-normed space.

On the pillars of Functional Analysis

Many authors consider that the main pillars of Functional Analysis are the Hahn–Banach Theorem, the Uniform Boundedness Principle and the Open Mapping Principle. The first one is derived from Zorn’s Lemma, while the latter two usually are obtained from Baire’s Category Theorem. In this paper we show that these three pillars should be either just two or at least eight, since the Uniform Boundedness Principle, the Open Mapping Principle and another five theorems are equivalent, as we show in a very elemental way. Since one can give an almost trivial proof of the Uniform Boundedness Principle that does not require the Baire’s theorem, we conclude that this is also the case for the other equivalent theorems that, in this way, are simultaneously proved in a simple, brief and concise way that sheds light on their nature.

Banach-Steinhaus Theorem for the Space P of All Primitives of Henstock-Kurzweil Integrable Functions

In this paper, it is shown how the Banach-Steinhaus theorem for the space P of all primitives of Henstock-Kurzweil integrable functions on a closed bounded interval, equipped with the uniform norm, can follow from the Banach-Steinhaus theorem for the Denjoy space by applying the classical Hahn-Banach theorem and Riesz representation theorem.

English

The space $${\cal H}{\cal K}$$ of Henstock-Kurzweil integrable functions on [a, b] is the uncountable union of Frechet spaces $${\cal H}{\cal K}$$ (X). In this paper, on each Frechet space $${\cal H}{\cal K}$$ (X), an F-norm is defined for a continuous linear operator. Hence, many important results in functional analysis, like the Banach-Steinhaus theorem, the open mapping theorem and the closed graph theorem, hold for the $${\cal H}{\cal K}$$ (X) space. It is known that every control-convergent sequence in the $${\cal H}{\cal K}$$ space always belongs to a $${\cal H}{\cal K}$$ (X) space for some X. We illustrate how to apply results for Frechet spaces $${\cal H}{\cal K}$$ (X) to control-convergent sequences in the $${\cal H}{\cal K}$$ space. Examples of compact linear operators are given. Existence of solutions to linear and Hammerstein integral equations is proved.

Convergence structures and locally solid topologies on vector lattices of operators

For vector lattices E and F, where F is Dedekind complete and supplied with a locally solid topology, we introduce the corresponding locally solid absolute strong operator topology on the order bounded operators {mathscr{L}}_{mathrm{ob}}(E,F) from E into F. Using this, it follows that {mathscr{L}}_{mathrm{ob}}(E,F) admits a Hausdorff uo-Lebesgue topology whenever F does. For each of order convergence, unbounded order convergence, and—when applicable—convergence in the Hausdorff uo-Lebesgue topology, there are both a uniform and a strong convergence structure on {mathscr{L}}_{mathrm{ob}}(E,F). Of the six conceivable inclusions within these three pairs, only one is generally valid. On the orthomorphisms of a Dedekind complete vector lattice, however, five are generally valid, and the sixth is valid for order bounded nets. The latter condition is redundant in the case of sequences of orthomorphisms, as a consequence of a uniform order boundedness principle for orthomorphisms that we establish. We furthermore show that, in contrast to general order bounded operators, orthomorphisms preserve not only order convergence of nets, but unbounded order convergence and—when applicable—convergence in the Hausdorff uo-Lebesgue topology as well.

The uniform boundedness theorem in b-Banach space

The uniform boundedness theorem in b-Banach space

On sharpness of error bounds for multivariate neural network approximation

Single hidden layer feedforward neural networks can represent multivariate functions that are sums of ridge functions. These ridge functions are defined via an activation function and customizable weights. The paper deals with best non-linear approximation by such sums of ridge functions. Error bounds are presented in terms of moduli of smoothness. The main focus, however, is to prove that the bounds are best possible. To this end, counterexamples are constructed with a non-linear, quantitative extension of the uniform boundedness principle. They show sharpness with respect to Lipschitz classes for the logistic activation function and for certain piecewise polynomial activation functions. The paper is based on univariate results in Goebbels (Res Math 75(3):1–35, 2020. https://rdcu.be/b5mKH)

Fuzzy bounded operators with application to Radon transform

Abstract This paper is focused on developing the means to extend the range of application of the inverse Radon transform by enlarging the domain of definition of the Radon operator, namely from a specific Banach space to a more general fuzzy normed linear space. This is done by studying different types of fuzzy bounded linear operators acting between fuzzy normed linear spaces. The motivation for considering this type of spaces comes from the existence of an equivalence between the probabilistic metric spaces and fuzzy metric spaces, in particular fuzzy normed linear spaces. We mention that many notions and results belonging to classical metric spaces could also be found in this general context. Moreover, this setup allows to develop applications as diverse as: image processing, data compression, signal processing, computer graphics, etc. The class of operators that best fits the intended purpose is the class of strongly fuzzy bounded linear operators. The main results about this family of operators use the fact that the space of such operators becomes a normed algebra. An extension of the classical norm of a bounded linear operator between two normed spaces to the norm of strongly fuzzy bounded linear operators acting between fuzzy normed linear spaces is proved. A version of the classical Banach-Steinhaus theorem for strongly fuzzy bounded linear operators is given. A sufficient condition for the limit of a sequence of strongly fuzzy bounded linear operators to be strongly fuzzy bounded is shown. The adjoint operator of a strongly fuzzy bounded linear operator is a classic bounded linear operator. The class of neighborhood fuzzy bounded linear operators are studied as well, being established connections with two other classes of operator, namely the class of fuzzy bounded linear operators and strongly fuzzy bounded linear operators.

Filter-dependent versions of the uniform boundedness principle

For every filter F on N, we introduce and study corresponding uniformF-boundedness principles for locally convex topological vector spaces. These principles generalise the classical uniform boundedness principles for sequences of continuous linear maps by coinciding with these principles when the filter F equals the Fréchet filter of cofinite subsets of N. We determine combinatorial properties for the filter F which ensure that these uniform F-boundedness principles hold for every Fréchet space. Furthermore, for several types of Fréchet spaces, we also isolate properties of F that are necessary for the validity of these uniform F-boundedness principles. For every infinite-dimensional Banach space X, we obtain in this way exact combinatorial characterisations of those filters F for which the corresponding uniform F-boundedness principles hold true for X.