Generalized M-norms on ordered normed spaces

I. Tzschichholtz; M. R. Weber

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Displaying similar documents to “Generalized M-norms on ordered normed spaces”

Modifications of the double arrow space and related Banach spaces C(K)

Witold Marciszewski (2008)

Studia Mathematica

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We consider the class of compact spaces $K_{A}$ which are modifications of the well known double arrow space. The space $K_{A}$ is obtained from a closed subset K of the unit interval [0,1] by “splitting” points from a subset A ⊂ K. The class of all such spaces coincides with the class of separable linearly ordered compact spaces. We prove some results on the topological classification of $K_{A}$ spaces and on the isomorphic classification of the Banach spaces $C (K_{A})$ .

$L_{1}$ contains every two-dimensional normed space

David Yost (1988)

Annales Polonici Mathematici

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Chromatic number of the product of graphs, graph homomorphisms, antichains and cofinal subsets of posets without AC

Amitayu Banerjee, Zalán Gyenis (2021)

Commentationes Mathematicae Universitatis Carolinae

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In set theory without the axiom of choice (AC), we observe new relations of the following statements with weak choice principles. $\circ$ If in a partially ordered set, all chains are finite and all antichains are countable, then the set is countable. $\circ$ If in a partially ordered set, all chains are finite and all antichains have size $ℵ_{α}$ , then the set has size $ℵ_{α}$ for any regular $ℵ_{α}$ . $\circ$ Every partially ordered set without a maximal element has two disjoint cofinal sub sets – CS. $\circ$ Every partially...

A theorem on isotropic spaces

Félix Cabello Sánchez (1999)

Studia Mathematica

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Let X be a normed space and $G_{F} (X)$ the group of all linear surjective isometries of X that are finite-dimensional perturbations of the identity. We prove that if $G_{F} (X)$ acts transitively on the unit sphere then X must be an inner product space.

Specialization to the tangent cone and Whitney equisingularity

Arturo Giles Flores (2013)

Bulletin de la Société Mathématique de France

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Let $(X, 0)$ be a reduced, equidimensional germ of an analytic singularity with reduced tangent cone $(C_{X, 0}, 0)$ . We prove that the absence of exceptional cones is a necessary and sufficient condition for the smooth part $𝔛^{0}$ of the specialization to the tangent cone $ϕ : 𝔛 \to ℂ$ to satisfy Whitney’s conditions along the parameter axis $Y$ . This result is a first step in generalizing to higher dimensions Lê and Teissier’s result for hypersurfaces of $ℂ^{3}$ which establishes the Whitney equisingularity of $X$ and its tangent...

On generalized derivations of partially ordered sets

Ahmed Y. Abdelwanis, Abdelkarim Boua (2019)

Communications in Mathematics

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Let $P$ be a poset and $d$ be a derivation on $P$ . In this research, the notion of generalized $d$ -derivation on partially ordered sets is presented and studied. Several characterization theorems on generalized $d$ -derivations are introduced. The properties of the fixed points based on the generalized $d$ -derivations are examined. The properties of ideals and operations related with generalized $d$ -derivations are studied.

On area and side lengths of triangles in normed planes

Gennadiy Averkov, Horst Martini (2009)

Colloquium Mathematicae

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Let $ℳ^{d}$ be a d-dimensional normed space with norm ||·|| and let B be the unit ball in $ℳ^{d}$ . Let us fix a Lebesgue measure $V_{B}$ in $ℳ^{d}$ with $V_{B} (B) = 1$ . This measure will play the role of the volume in $ℳ^{d}$ . We consider an arbitrary simplex T in $ℳ^{d}$ with prescribed edge lengths. For the case d = 2, sharp upper and lower bounds of $V_{B} (T)$ are determined. For d ≥ 3 it is noticed that the tight lower bound of $V_{B} (T)$ is zero.

Counting linearly ordered spaces

Gerald Kuba (2014)

Colloquium Mathematicae

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For a transfinite cardinal κ and i ∈ 0,1,2 let $ℒ_{i} (κ)$ be the class of all linearly ordered spaces X of size κ such that X is totally disconnected when i = 0, the topology of X is generated by a dense linear ordering of X when i = 1, and X is compact when i = 2. Thus every space in ℒ₁(κ) ∩ ℒ₂(κ) is connected and hence ℒ₁(κ) ∩ ℒ₂(κ) = ∅ if $κ < 2^{ℵ ₀}$ , and ℒ₀(κ) ∩ ℒ₁(κ) ∩ ℒ₂(κ) = ∅ for arbitrary κ. All spaces in ℒ₁(ℵ₀) are homeomorphic, while ℒ₂(ℵ₀) contains precisely ℵ₁ spaces up to homeomorphism. The...

Spaces of continuous step functions over LOTS

Raushan Z. Buzyakova (2006)

Fundamenta Mathematicae

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We investigate spaces $C_{p} (\cdot, n)$ over LOTS (linearly ordered topological spaces). We find natural necessary conditions for linear Lindelöfness of $C_{p} (\cdot, n)$ over LOTS. We also characterize countably compact LOTS whose $C_{p} (\cdot, n)$ is linearly Lindelöf for each n. Both the necessary conditions and the characterization are given in terms of the topology of the Dedekind completion of a LOTS.

On linear maps leaving invariant the copositive/completely positive cones

Sachindranath Jayaraman, Vatsalkumar N. Mer (2024)

Czechoslovak Mathematical Journal

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The objective of this manuscript is to investigate the structure of linear maps on the space of real symmetric matrices $𝒮^{n}$ that leave invariant the closed convex cones of copositive and completely positive matrices ( ${COP}_{n}$ and ${CP}_{n}$ ). A description of an invertible linear map on $𝒮^{n}$ such that $L ({CP}_{n}) \subset C P_{n}$ is obtained in terms of semipositive maps over the positive semidefinite cone $𝒮_{+}^{n}$ and the cone of symmetric nonnegative matrices $𝒩_{+}^{n}$ for $n \leq 4$ , with specific calculations for $n = 2$ . Preserver properties of the Lyapunov...

On Some Properties of Separately Increasing Functions from [0,1]ⁿ into a Banach Space

Artur Michalak (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

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We say that a function f from [0,1] to a Banach space X is increasing with respect to E ⊂ X* if x* ∘ f is increasing for every x* ∈ E. A function $f : {[0, 1]}^{m} \to X$ is separately increasing if it is increasing in each variable separately. We show that if X is a Banach space that does not contain any isomorphic copy of c₀ or such that X* is separable, then for every separately increasing function $f : {[0, 1]}^{m} \to X$ with respect to any norming subset there exists a separately increasing function $g : {[0, 1]}^{m} \to ℝ$ such that the sets of...

On the H-property and rotundity of Cesàro direct sums of Banach spaces

Saard Youyen, Suthep Suantai (2008)

Banach Center Publications

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In this paper, we define the direct sum ${(⨁_{i = 1}^{n} X_{i})}_{c e s_{p}}$ of Banach spaces X₁,X₂,..., and Xₙ and consider it equipped with the Cesàro p-norm when 1 ≤ p < ∞. We show that ${(⨁_{i = 1}^{n} X_{i})}_{c e s_{p}}$ has the H-property if and only if each $X_{i}$ has the H-property, and ${(⨁_{i = 1}^{n} X_{i})}_{c e s_{p}}$ has the Schur property if and only if each $X_{i}$ has the Schur property. Moreover, we also show that ${(⨁_{i = 1}^{n} X_{i})}_{c e s_{p}}$ is rotund if and only if each $X_{i}$ is rotund.

Pisier's inequality revisited

Tuomas Hytönen, Assaf Naor (2013)

Studia Mathematica

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Given a Banach space X, for n ∈ ℕ and p ∈ (1,∞) we investigate the smallest constant ∈ (0,∞) for which every n-tuple of functions f₁,...,fₙ: -1,1ⁿ → X satisfies $\int_{- 1, 1 ⁿ} | | \sum_{j = 1}^{n} \partial_{j} f_{j} {(ε) | |}^{p} d μ (ε) \leq^{p} \int_{- 1, 1 ⁿ} \int_{- 1, 1 ⁿ} | | \sum_{j = 1}^{n} δ_{j} Δ f_{j} (ε) {| |}^{p} d μ (ε) d μ (δ)$ , where μ is the uniform probability measure on the discrete hypercube -1,1ⁿ, and ${\partial_{j}}_{j = 1}^{n}$ and $Δ = \sum_{j = 1}^{n} \partial_{j}$ are the hypercube partial derivatives and the hypercube Laplacian, respectively. Denoting this constant by $ⁿ_{p} (X)$ , we show that $ⁿ_{p} (X) \leq \sum_{k = 1}^{n} 1 / k$ for every Banach space (X,||·||). This extends the classical Pisier inequality, which corresponds to the special...

Sequentially Right Banach spaces of order $p$

Mahdi Dehghani, Mohammad B. Dehghani, Mohammad S. Moshtaghioun (2020)

Commentationes Mathematicae Universitatis Carolinae

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We introduce and study two new classes of Banach spaces, the so-called sequentially Right Banach spaces of order $p$ , and those defined by the dual property, the sequentially Right $^{*}$ Banach spaces of order $p$ for $1 \leq p \leq \infty$ . These classes of Banach spaces are characterized by the notions of $L_{p}$ -limited sets in the corresponding dual space and $R_{p}^{*}$ subsets of the involved Banach space, respectively. In particular, we investigate whether the injective tensor product of a Banach space $X$ and a reflexive Banach...

On the Aronszajn property for integral equations in Banach space

Stanisław Szufla (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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For the integral equation (1) below we prove the existence on an interval $J=[0,a]$ of a solution $x$ with values in a Banach space $E$ , belonging to the class $L^{p}(J,E)$ , $p>1$ . Further, the set of solutions is shown to be a compact one in the sense of Aronszajn.

Lanczos-like algorithm for the time-ordered exponential: The $*$ -inverse problem

Pierre-Louis Giscard, Stefano Pozza (2020)

Applications of Mathematics

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The time-ordered exponential of a time-dependent matrix $𝖠 (t)$ is defined as the function of $𝖠 (t)$ that solves the first-order system of coupled linear differential equations with non-constant coefficients encoded in $𝖠 (t)$ . The authors have recently proposed the first Lanczos-like algorithm capable of evaluating this function. This algorithm relies on inverses of time-dependent functions with respect to a non-commutative convolution-like product, denoted by $*$ . Yet, the existence of such inverses,...

Reflexivity and approximate fixed points

Eva Matoušková, Simeon Reich (2003)

Studia Mathematica

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A Banach space X is reflexive if and only if every bounded sequence xₙ in X contains a norm attaining subsequence. This means that it contains a subsequence $x_{n_{k}}$ for which $s u p_{f \in S_{X *}} l i m s u p_{k \to \infty} f (x_{n_{k}})$ is attained at some f in the dual unit sphere $S_{X *}$ . A Banach space X is not reflexive if and only if it contains a normalized sequence xₙ with the property that for every $f \in S_{X *}$ , there exists $g \in S_{X *}$ such that $l i m s u p_{n \to \infty} f (x ₙ) < l i m i n f_{n \to \infty} g (x ₙ)$ . Combining this with a result of Shafrir, we conclude that every infinite-dimensional Banach space contains an unbounded...

C*-algebras have a quantitative version of Pełczyński's property (V)

Hana Krulišová (2017)

Czechoslovak Mathematical Journal

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A Banach space $X$ has Pełczyński’s property (V) if for every Banach space $Y$ every unconditionally converging operator $T : X \to Y$ is weakly compact. H. Pfitzner proved that $C^{*}$ -algebras have Pełczyński’s property (V). In the preprint (Krulišová, (2015)) the author explores possible quantifications of the property (V) and shows that $C (K)$ spaces for a compact Hausdorff space $K$ enjoy a quantitative version of the property (V). In this paper we generalize this result by quantifying Pfitzner’s theorem. Moreover,...

$ℓ_{\infty}$ -sums and the Banach space $ℓ_{\infty} / c ₀$

Christina Brech, Piotr Koszmider (2014)

Fundamenta Mathematicae

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This paper is concerned with the isomorphic structure of the Banach space $ℓ_{\infty} / c ₀$ and how it depends on combinatorial tools whose existence is consistent with but not provable from the usual axioms of ZFC. Our main global result is that it is consistent that $ℓ_{\infty} / c ₀$ does not have an orthogonal $ℓ_{\infty}$ -decomposition, that is, it is not of the form $ℓ_{\infty} (X)$ for any Banach space X. The main local result is that it is consistent that $ℓ_{\infty} (c ₀ ())$ does not embed isomorphically into $ℓ_{\infty} / c ₀$ , where is the cardinality of the continuum,...

The extension of the Krein-Šmulian theorem for order-continuous Banach lattices

Antonio S. Granero, Marcos Sánchez (2008)

Banach Center Publications

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If X is a Banach space and C ⊂ X a convex subset, for x** ∈ X** and A ⊂ X** let d(x**,C) = inf||x**-x||: x ∈ C be the distance from x** to C and d̂(A,C) = supd(a,C): a ∈ A. Among other things, we prove that if X is an order-continuous Banach lattice and K is a w*-compact subset of X** we have: (i) $d ̂ ({\bar{c o}}^{w *} (K), X) \leq 2 d ̂ (K, X)$ and, if K ∩ X is w*-dense in K, then $d ̂ ({\bar{c o}}^{w *} (K), X) = d ̂ (K, X)$ ; (ii) if X fails to have a copy of ℓ₁(ℵ₁), then $d ̂ ({\bar{c o}}^{w *} (K), X) = d ̂ (K, X)$ ; (iii) if X has a 1-symmetric basis, then $d ̂ ({\bar{c o}}^{w *} (K), X) = d ̂ (K, X)$ .

On some properties of generalized Marcinkiewicz spaces

Evgeniy Pustylnik (2001)

Studia Mathematica

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We give a full solution of the following problems concerning the spaces $M_{φ} (X ⃗)$ : (i) to what extent two functions φ and ψ should be different in order to ensure that $M_{φ} (X ⃗) \neq M_{ψ} (X ⃗)$ for any nontrivial Banach couple X⃗; (ii) when an embedding $M_{φ} (X ⃗) ⊊ M_{ψ} (X ⃗)$ can (or cannot) be dense; (iii) which Banach space can be regarded as an $M_{φ} (X ⃗)$ -space for some (unknown beforehand) Banach couple X⃗.

Maximal independent sets, variants of chain/antichain principle and cofinal subsets without AC

Amitayu Banerjee (2023)

Commentationes Mathematicae Universitatis Carolinae

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In set theory without the axiom of choice (AC), we observe new relations of the following statements with weak choice principles. $\circ$ $𝒫_{lf, c}$ (Every locally finite connected graph has a maximal independent set). $\circ$ $𝒫_{lc, c}$ (Every locally countable connected graph has a maximal independent set). $\circ$ CAC $_{1}^{ℵ_{α}}$ (If in a partially ordered set all antichains are finite and all chains have size $ℵ_{α}$ , then the set has size $ℵ_{α}$ ) if $ℵ_{α}$ is regular. $\circ$ CWF (Every partially ordered set has a...