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Journal: Annales Mathematicae Silesianae

On various types of uniform rotundities

Tulsi Dass Narang , Sumit Chandok , Sahil Gupta

Language: EN | Published: 04-03-2025 | Abstract | pp. 281-318

In this study, we conduct a literature review on normed linear spaces whose strengths are between rotundity and uniform rotundity. In this discourse, we also explore inter-relationships and juxtapositions between the subjects under consideration. There has been some discussion on the extent to which the geometry of the factor spaces has an impact on the geometry of the product spaces, as well as the degree to which the quotient spaces and subspaces inherit the geometry of the space itself. A comprehensive review has been conducted on the applications of most of these rotundities to some fields within the realm of approximation theory. In addition, some open problems are enumerated in the paper.

2025-03-04



Journal: Annales Mathematicae Silesianae

Characterizations of rotundity and smoothness by approximate orthogonalities

Tomasz Stypuła , Paweł Wójcik

Language: EN | Published: 23-09-2016 | Abstract | pp. 193-201

In this paper we consider the approximate orthogonalities in real normed spaces. Using the notion of approximate orthogonalities in real normed spaces, we provide some new characterizations of rotundity and smoothness of dual spaces.

2016-09-23



Journal: Annales Mathematicae Silesianae

Connections between the completion of normed spaces over non-archimedean fields and the stability of the Cauchy equation

Jens Schwaiger

Language: EN | Published: 08-05-2020 | Abstract | pp. 151-163

In [12] a close connection between stability results for the Cauchy equation and the completion of a normed space over the rationals endowed with the usual absolute value has been investigated. Here similar results are presented when the valuation of the rationals is a p-adic valuation. Moreover a result by Zygfryd Kominek ([5]) on the stability of the Pexider equation is
formulated and proved in the context of Banach spaces over the field of p-adic numbers.

2020-05-08



Journal: Annales Mathematicae Silesianae

The d'Alembert and Lobaczevski difference operators in ????_λ spaces

Maciej Przybyła

Language: EN | Published: 29-09-2006 | Abstract | pp. 7-17

Let X be a linear normed space, λ≥0, n∈ℕ. Let ????_λ⁽ⁿ⁾ be a set defined by
????_λ⁽ⁿ⁾ = {g: Xⁿ→ℂ |  |g(x^→)| ≤ M_g · e^{λΣ_k=1n‖x_k‖}, x^→∈Xⁿ},
where M_g is a constant depending on g. Moreover for all g∈????_λ⁽ⁿ⁾ we define
‖g‖ := sup_x^→∈Xⁿ {e^{-λΣ_k=1n‖x_k‖} · |g(x^→)|}.
In the paper norms of the d'Alembert and Lobaczevski difference operators in the ????_λⁿ spaces are calculated (their Pexider type generalizations are also considered). Moreover it is proved that if f: X→ℂ is a function such that A(f)∈????_λ⁽²⁾, where A is the d'Alembert difference operator, then f∈????_λ or A(f) = 0.

2006-09-29



Journal: Annales Mathematicae Silesianae

Über die Stetigkeit einer Funktion von zwei Veränderlichen unter Monotonie-Bedingungen

Manfred Krtscha , Peter Volkmann

Language: EN | Published: 30-09-1991 | Abstract | pp. 18-27

Let T, E, F be topological spaces, and let E, F also be preordered with some compatibility between topology and order stucture. Theorem: If f(t,x): T×E→F is separately continuous with respect to its variables and monotone non-decreasing with respect to x, then f is continuous. The case of ordered normed spaces E, F is discussed in detail.

1991-09-30



Journal: Annales Mathematicae Silesianae

Classification of Lipschitz derivatives in terms of semicontinuity and the Baire limit functions

Oleksandr V. Maslyuchenko , Ziemowit M. Wójcicki

Language: EN | Published: 24-02-2026 | Abstract

We introduce the generalized notion of semicontinuity of a function defined on a topological space and derive the useful classification of the socalled Lipschitz derivatives of functions defined on a metric space. Secondly, we investigate some connections of the Lipschitz derivatives defined on normed spaces to the Fréchet derivative and relations between little, big and local Lipschitz derivatives (denoted by lip f, Lip f and Lip f respectively) in terms of Baire limit functions. In particular, we prove that lip f is F_σ-lower, Lip f is F_σ-upper, Lip f is upper semicontinuous. Moreover, for a function f defined on an open or convex subset of a normed space, the upper Baire limit function of functions lip f and Lip f are equal to Lip f.

2026-02-24



Journal: Annales Mathematicae Silesianae

Strong m-convexity of set-valued functions

Teodoro Lara , Nelson Merentes , Roy Quintero , Edgar Rosales

Language: EN | Published: 07-02-2023 | Abstract | pp. 82-94

In this research we introduce the concept of strong m-convexity for set-valued functions defined on m-convex subsets of real linear normed spaces, a variety of properties and examples of these functions are shown, an inclusion of Jensen type is also exhibited.

2023-02-07