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We compute the numerical index of the two-dimensional real $ L_p $ L p space for $ \frac {6}{5}\leqslant p \leqslant 1+\alpha _0 $ 6 5 ⩽ p ⩽ 1 + α 0 and $ \alpha _1\leqslant p\leqslant 6 $ α 1 ⩽ p ⩽ 6 , where $ \alpha _0 $ α 0 is the root of $ f(x)=1+x^{-2}-(x^{-\frac {1}{x}}+x^{\frac {1}{x}}) $ f (x) = 1 + x − 2 − (x − 1 x + x 1 x ) and $ \frac {1}{1+\alpha _0}+\frac {1}{\alpha _1}=1 $ 1 1 + α 0 + 1 α 1 = 1. This, together with the previous results in Merí and Quero [On the numerical index of absolute symmetric norms on the plane. Linear Multilinear Algebra. 2021;69(5):971–979] and Monika and Zheng [The numerical index of $ \ell _p^2 $ ℓ p 2 . Linear Multilinear Algebra. 2022;1–6. Published online DOI:10.1080/03081087.2022.2043818], gives the numerical index of the two-dimensional real $ L_p $ L p space for $ \frac {6}{5}\leqslant p \leqslant 6 $ 6 5 ⩽ p ⩽ 6. [ABSTRACT FROM AUTHOR]

In this work, it is assumed that the norm over bicomplex numbers is the hyperbolic (픻 -valued) norm. In this paper, we provide an overview of bicomplex Lebesgue spaces and investigate some of their geometric properties, including 픹 ⁢ ℂ -convexity, 픹 ⁢ ℂ -strict convexity, and 픹 ⁢ ℂ -uniform convexity. Moreover, the basic inequalities such as 픻 -Hölder's inequality and 픻 -Minkowski inequality for bicomplex Lebesgue spaces are presented, used to show geometric properties. [ABSTRACT FROM AUTHOR]

In this note, we are mainly interested in orthogonality, including Birkhoff orthogonality, isosceles orthogonality, Pythagorean orthogonality and approximate Birkhoff orthogonality ⊥ B ϵ . First, we show the relation between linear functionals and hyperplanes by means of approximate Birkhoff orthogonality. Second, we establish the sufficient conditions for Birkhoff orthogonality and approximate Birkhoff orthogonality. Then some sufficient and necessary conditions for isosceles orthogonality and Pythagorean orthogonality of bounded operators are shown. In addition, we characterize the inner product space in terms of functional orthogonality. [ABSTRACT FROM AUTHOR]

The aim of this paper is to discuss some results concerning the demiclosedness principle of generalized, nonexpansive mappings in uniformly convex spaces. Further, we present some new fixed-point theorems for generalized nonexpansive mappings in different settings of Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in studying Bishop–Phelps–Bollobás type properties related to the denseness of the operators which attain their numerical radius. We prove that every Banach space with a micro-transitive norm and the second numerical index strictly positive satisfies the Bishop–Phelps–Bollobás point property, and we see that the one-dimensional space is the only one with both the numerical index 1 and the Bishop–Phelps–Bollobás point property. We also consider two weaker properties L $ {}_{p,p} $ p , p -nu and L $ {}_{o,o} $ o , o -nu, the local versions of Bishop–Phelps–Bollobás point and operator properties respectively, where the η which appears in their definition does not depend just on $ \varepsilon > 0 $ ε > 0 but also on a state $ (x, x^*) $ (x , x ∗) or on a numerical radius one operator T. We address the relation between the L $ {}_{p,p} $ p , p -nu and the strong subdifferentiability of the norm of the space X. We show that finite dimensional spaces and $ c_0 $ c 0 are examples of Banach spaces satisfying the L $ {}_{p,p} $ p , p -nu, and we exhibit an example of a Banach space with a strongly subdifferentiable norm failing it. We finish the paper by showing that finite dimensional spaces satisfy the L $ {}_{o,o} $ o , o -nu and that, if X has a strictly positive numerical index and has the approximation property, this property is equivalent to finite dimensionality. [ABSTRACT FROM AUTHOR]

Let A be a unital C ∗ -algebra with unit 1 A and let a ∈ A be a positive and invertible element. Suppose that S (A) is the set of all states on A and let S a (A) = f f (a) : f ∈ S (A) , f (a) ≠ 0. The norm ‖ x ‖ a for every x ∈ A is defined by ‖ x ‖ a = sup φ ∈ S a (A) φ (x ∗ a x). In this paper, we aim to investigate the notion of Birkhoff–James orthogonality with respect to the norm ‖ · ‖ a in A , namely a-Birkhoff–James orthogonality. The characterization of a-Birkhoff–James orthogonality in A by means of the elements of generalized state space S a (A) is provided. As an application, a characterization for the best approximation to elements of A in a subspace B with respect to ‖ · ‖ a is obtained. Moreover, a formula for the distance of an element of A to the subspace B = C 1 A is given. We also study the strong version of a-Birkhoff–James orthogonality in A. The classes of C ∗ -algebras in which these two types orthogonality relationships coincide are described. In particular, we prove that the condition of the equivalence between the strong a-Birkhoff–James orthogonality and A -valued inner product orthogonality in A implies that the center of A is trivial. Finally, we show that if the (strong) a-Birkhoff–James orthogonality is right-additive (left-additive) in A , then the center of A is trivial. [ABSTRACT FROM AUTHOR]

This article is devoted to introduce a new geometric constant called Dehghan–Rooin constant, which quantifies the difference between angular and skew angular distances in Banach spaces. We quantify the characterization of uniform non-squareness in terms of Dehghan–Rooin constant. The relationships between Dehghan–Rooin constant and uniform convexity, Dehghan-Rooin constant and uniform smoothness are also studied. Moreover, some new sufficient conditions for uniform normal structure are also established in terms of Dehghan–Rooin constant. [ABSTRACT FROM AUTHOR]

In this article, with simple and short proofs without applying fixed point theorems, some hyperstability results corresponding to the functional equations of Cauchy and Jensen are presented in 2-normed spaces. We also obtain some results on hyperstability for the general linear functional equation f (a x + b y) = A f (x) + B f (y) + C . [ABSTRACT FROM AUTHOR]

In this paper, we shall consider a new constant D W B (X) which is the Dunkl–Williams constant related to Birkhoff orthogonality, and a constant D W B p (X) that is a generalization of D W B (X) . Interestingly, the upper bounds of D W B (X) and the Dunkl–Williams constant are different. The connections between these two constants and other well-known constants are exhibited. Some characterizations of Hilbert space and uniformly non-square Banach space in terms of these two constants are established. Furthermore, we also give a characterization of the Radon plane with an affine regular hexagonal unit sphere and calculate the value of D W B p (l ∞ - l 1) . [ABSTRACT FROM AUTHOR]

Given an infinite matrix M = (m nk) , we study a family of sequence spaces ℓ M p associated with it. When equipped with a suitable norm ‖ · ‖ M , p , we prove some basic properties of the Banach spaces of sequences (ℓ M p , ‖ · ‖ M , p) . In particular, we show that such spaces are separable and strictly/uniformly convex for a considerably large class of infinite matrices M for all p > 1 . A special attention is given to the identification of the dual space (ℓ M p) ∗ . Building on the earlier works of Bennett and Jägers, we extend and apply some classical factorization results to the sequence spaces ℓ M p . [ABSTRACT FROM AUTHOR]

The aim of the paper is to discuss and clarify some concepts of the geometric theory of normed spaces. We mainly intend to present recent results concerning the concept of smoothness of normed spaces in connection with the concepts of the strict and uniform convexity of those spaces. [ABSTRACT FROM AUTHOR]

We study two constants g 1 (X) and J 1 (X) introduced by Fu et al. (Symmetry 13(6):951, 2021), present new characterizations of them, clarify detailed relations of these constants to James and Schäffer constants as well as the relation between J 1 (X) and A 2 (X) defined by Baronti et al. (J Math Anal Appl 252(1):124–146, 2000). We also pose several problems. [ABSTRACT FROM AUTHOR]

The aim of the present article is to extend the Stone-Weierstrass theorem to functions ranging in a lattice normed space and admitting order rather than topological approximation. We proceed with the machinery of Boolean-valued transfer from lattice normed space to normed space. [ABSTRACT FROM AUTHOR]

Angular equivalence of norms, introduced by Kikianty and Sinnamon (2017), is a notion of norm equivalence that is more attuned to the geometry of the norms. For certain geometrical properties and two angularly equivalent norms, it is the case that if one of the norms has a property, then so does the other. In this paper, we show further results in this direction, namely angularly equivalent norms share the property of non-squareness; and that the exposed points of the unit balls are in the same direction, under the condition that these points are assumed to be smooth with respect to both norms. A discussion on (the angular equivalence of) the dual norms of angularly equivalent norms is also given, giving a partial answer to an open problem as stated in the paper by Kikianty and Sinnamon (2017). [ABSTRACT FROM AUTHOR]

Suppose X and Y are Banach spaces, K is a compact Hausdorff space, Σ is the σ-algebra of Borel subsets of K , C (K, X) is the Banach space of all continuous X-valued functions (with the supremum norm), and T : C (K, X) → Y is a strongly bounded operator with representing measure m : Σ → L(X, Y). We show that if T is a strongly bounded operator and : B(K, X) → Y is its extension, then T* is pseudo weakly compact (resp. limited completely continuous, limited p-convergent, 1 ≤ p < ∞) if and only if has the same property. [ABSTRACT FROM AUTHOR]

In this work, we give some criteria of the weakly compact sets and a representation theorem of Riesz's type in Musielak sequence spaces using the ideas and techniques of sequence spaces and Musielak function. Finally, as an immediate consequence of the criteria considered in this paper, the criteria of the weakly compact sets of Orlicz sequence spaces are deduced. [ABSTRACT FROM AUTHOR]

In this article, we prove the existence of the best proximity point for the class of nonself generalized pseudo-contractive and Lipschitzian mappings. Also, we approximate the best proximity point through the proposed Ishikawa's iteration process for the case of nonself-mappings. Finally, we provide an example to illustrate our main result. [ABSTRACT FROM AUTHOR]

Let X , Y be real, infinite-dimensional Banach spaces. Let ℒ ⁢ (X , Y) be the space of bounded operators. An important aspect of understanding differentiability of the operator norm at A ∈ ℒ ⁢ (X , Y) is to estimate the limit (which always exists) lim t → 0 + ⁡ ∥ A + t ⁢ B ∥ - ∥ A ∥ t for ⁢ B ∈ ℒ ⁢ (X , Y) , using the values of B on the state space S A = { τ ∈ ℒ ⁢ (X , Y) ∗ : τ ⁢ (A) = ∥ A ∥ , ∥ τ ∥ = 1 }. In this paper, we give several examples of Banach spaces, including the ℓ p spaces (for 1 < p < ∞ ) where a more tangible estimate is possible, under additional hypotheses on A. We also use the notion of norm-weak upper-semi-continuity (usc, for short) of the preduality map to achieve this. Our results also show that the operator subdifferential limit is related to the corresponding subdifferential limit of the vectors in the range space, when A ∗ ∗ attains its norm. [ABSTRACT FROM AUTHOR]

Gähler (Untersuchungen über verallgemeinerte m-metrische Räume. I". German. Math Nachr 40, 165–189, 1969) investigated m-metric spaces and in particular m-normed spaces over the field of reals. Here the existence of m-norms will be investigated and this in the more general setting of vector spaces over arbitrary non-trivial valued fields. [ABSTRACT FROM AUTHOR]

For the well developed notion of approximate Birkhoff-James orthogonality, in a real or complex normed linear space, we formulate a new characterization. It can be derived from other, already known, characterizations as well as obtained in a more elementary and direct way, on the basis of some simple inequalities for real convex functions. [ABSTRACT FROM AUTHOR]

We study in this paper a few remarkable properties of twisted sums Z(κ, X) of c0(κ) and a Banach space X. We first prove a representation theorem for such twisted sums from which we will obtain, among others, the following: (a) twisted sums of c0(κ) and c0(I) are either subspaces of ℓ∞(κ) or contain a complemented copy of c0(κ+); (b) under the hypothesis [p = c], when K is either a suitable Corson compact, a separable Rosenthal compact or a scattered compact of finite height, there is a twisted sum of c0 and C(K) that is not isomorphic to a space of continuous functions; (c) all twisted sums Z(κ, X) are isomorphically Lindenstrauss spaces when X is a Lindenstrauss space; (d) all twisted sums Z(κ, X) are isomorphically polyhedral when X is a polyhedral space with a σ-discrete boundary, which solves a problem of Castillo and Papini. [ABSTRACT FROM AUTHOR]

Given an open subset U of C n , a weight v on U and a complex Banach space F, let H v (U , F) denote the Banach space of all weighted holomorphic mappings f : U → F , under the weighted supremum norm f v : = sup v (z) f (z) : z ∈ U. We prove that the set of all mappings f ∈ H v (U , F) that attain their weighted supremum norms is norm dense in H v (U , F) , provided that the closed unit ball of the little weighted holomorphic space H v 0 (U , F) is compact-open dense in the closed unit ball of H v (U , F). We also prove a similar result for mappings f ∈ H v (U , F) such that vf has a relatively compact range. [ABSTRACT FROM AUTHOR]

The current best algorithms for the convex body chasing (CBC) problem in online algorithms use the notion of the Steiner point of a convex set. In particular, the algorithm that always moves to the Steiner point of the request set is O(d) competitive for nested CBC, and this is optimal among memoryless algorithms [Bubeck et al.: Chasing nested convex bodies nearly optimally. In: 31st Annual ACM–SIAM Symposium on Discrete Algorithms (Salt Lake City 2020), pp. 1496–1508. SIAM, Philadelphia (2020)]. A memoryless algorithm coincides with the notion of a selector in functional analysis. The Steiner point is noted for being Lipschitz with respect to the Hausdorff metric, and for achieving the minimal Lipschitz constant possible. It is natural to ask whether every selector with this Lipschitz property yields a competitive algorithm for nested CBC. We answer this question in the negative by exhibiting a selector that yields a non-competitive algorithm for nested CBC but is Lipschitz with respect to Hausdorff distance. Furthermore, we show that being Lipschitz with respect to an L p -type analog of the Hausdorff distance is sufficient to guarantee competitiveness if and only if p = 1 . [ABSTRACT FROM AUTHOR]

In the present paper, we introduce a new regular matrix B ~ and define the sequence space ℓ p ( B ~ ). We also examine some topological properties for this space and determine α - , β - , γ - duals. As well, we characterize some matrix classes and investigate some geometric properties such as uniform convexity, strict convexity and super-reflexivity. [ABSTRACT FROM AUTHOR]

Let Y be a uniformly convex space with power type p, and let (G , +) be an abelian group, δ , ε ≥ 0 , 0 < r < 1 . We first show a stability result for approximate isometries from an arbitrary Banach space into Y. This is a generalization of Dolinar' results for (δ , r) -isometries of Hilbert spaces and L p (1 < p < ∞ ) spaces. As a result, we prove that if a standard mapping F : G → Y satisfies d (u , F (G)) ≤ δ ∥ u ∥ r for every u ∈ Y and | ∥ F (x) − F (y) ∥ − ∥ F (x − y) ∥ | ≤ ε , x , y ∈ G , then there is an additive operator A : G → Y such that ∥ F (x) − A x ∥ = o (∥ F (x) ∥) as ∥ F (x) ∥ → ∞. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the reverse of the generalized triangle inequality of the second type in quasi-Banach spaces. More exactly, by using the concept of equivalent p-norms, we provide some necessary and sufficient conditions for n-tuples to satisfy the mentioned inequality. As applications, we improve some already known results and present some characterizations of p-Banach spaces among quasi-Banach spaces. In particular, we prove that a quasi-Banach space such as X is a p-Banach space if and only if for all (μ 1 , ⋯ , μ n) ∈ R n satisfying μ j > 0 for some j and μ i < 0 for all i ≠ j , the generalized triangle inequality of the second type ∑ i = 1 n ‖ x i ‖ p μ i ≤ ‖ ∑ i = 1 n x i ‖ p (x i ∈ X) holds only with the assumption μ j ≥ max i ∈ { 1 , ⋯ , n } \ { j } { 1 , | μ i | } . [ABSTRACT FROM AUTHOR]

We show that the algebra of cylinder functions in the Wasserstein Sobolev space H 1 , q (P p (X , d) , W p , d , m) generated by a finite and positive Borel measure m on the (p , d) -Wasserstein space (P p (X , d) , W p , d) on a complete and separable metric space (X , d) is dense in energy. As an application, we prove that, in case the underlying metric space is a separable Banach space B , then the Wasserstein Sobolev space is reflexive (resp. uniformly convex) if B is reflexive (resp. if the dual of B is uniformly convex). Finally, we also provide sufficient conditions for the validity of Clarkson's type inequalities in the Wasserstein Sobolev space. [ABSTRACT FROM AUTHOR]

We identify a norm-dense subspace of the dual of the sequence space l (p , ∞) , thus closing the existing gap in the literature. We based our approach on the notion of James orthogonality, absolutely continuous norms and on the uniform convexity and the uniform smoothness of the underlying subspaces. [ABSTRACT FROM AUTHOR]

We study the weak convergence of infinite products of convex combinations of operators in complete CAT(0) spaces. We provide a new approach to this problem by considering a constructive selection of convex combinations in CAT(0) spaces that does not depend on the order of the involved elements and retain continuity properties with respect to them. [ABSTRACT FROM AUTHOR]

A new orthogonality type in normed linear spaces, which is based on invariant inner products, is introduced. It is shown that this orthogonality has properties of existence, uniqueness, and homogeneity in general normed linear spaces. New characterizations of inner product spaces are obtained by studying properties of this orthogonality and its relation to other orthogonality types, including Birkhoff orthogonality, isosceles orthogonality, and Pythagorean orthogonality. Since this orthogonality is not additive in general, sufficient conditions for its local additivity are presented. Finally, a new geometric constant i(X) is introduced to characterize the difference between this orthogonality and isosceles orthogonality quantitatively. It is shown that 0 ≤ i (X) ≤ 1 holds in any normed linear space, i (X) = 0 iff the underlying space is an inner product space, and that the upper bound 1 is not attainable. [ABSTRACT FROM AUTHOR]

In this note we present two new positive answers to Tingley's problem in certain subspaces of function algebras. In the first result we prove that every surjective isometry between the unit spheres, S(A) and S(B), of two uniformly closed function algebras A and B on locally compact Hausdorff spaces can be extended to a surjective real linear isometry from A onto B. In a second part we study surjective isometries between the unit spheres of two abelian JB*-triples represented as spaces of continuous functions of the form where X is a (locally compact Hausdorff) principal -bundle and denotes the unit sphere of ℂ. We establish that every surjective isometry admits an extension to a surjective real linear isometry between these two abelian JB*-triples. [ABSTRACT FROM AUTHOR]

We consider a new type of cyclic (noncyclic) mappings, called diametrically relatively nonexpansive maps which contains properly the class of cyclic (noncyclic) relatively nonexpansive mappings. For such mappings we obtain existence results of best proximity points (pairs) in the framework of Busemann convex spaces and generalize the recent conclusions in this direction. We also present a characterization of proximal normal structure in term of best proximity points (pairs) for diametrically relatively nonexpansive mappings. [ABSTRACT FROM AUTHOR]

In this work, we study Bishop–Phelps cones (briefly, BP cones) given by an equation in Banach spaces. Due to the special form, these cones enjoy interesting properties. We show that nontrivial BP cones have given by an equation form a 'large family' in some sense in any Banach space and they can be used to characterize the strict convexity of the space. We obtain conditions for a convex cone to be included in or to be itself a BP cone given by an equation. Furthermore, we give an affirmative answer to the open question of whether these cones may possess a nonempty interior in infinite-dimensional spaces and introduce Lorentz cones in some classical Banach spaces of sequences as illustrations. We also obtain explicit representations for all BP cones given by an equation in some classical Banach spaces. Finally, we present some short applications in optimal control and approximation. [ABSTRACT FROM AUTHOR]

Let ( X , ⟨ ⋅ , ⋅ ⟩) be a Hilbert C ∗ -module over a C ∗ -algebra A and let S (A) be the set of states on A . In this paper, we first compute the norm derivative for nonzero elements x and y of X as follows: lim t → 0 + ‖ x + t y ‖ − ‖ x ‖ t = 1 ‖ x ‖ max { Re φ (⟨ x , y ⟩) : φ ∈ S (A) , φ (⟨ x , x ⟩) = ‖ x ‖ 2 }. We then apply it to characterize different concepts of orthogonality in X . In particular, we present a simpler proof of the classical characterization of Birkhoff–James orthogonality in Hilbert C ∗ -modules. Moreover, some generalized Daugavet equation in the C ∗ -algebra B (H) of all bounded linear operators acting on a Hilbert space H is solved. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study the quasi-H o ¨ lder mappings between unit spheres of p-normed spaces. The quasi-H o ¨ lder map is a natural generalization of the H o ¨ lder map, Lipschitz map, quasi-isometry and ε -isometry, etc. Concretely, we study the extension of a quasi-H o ¨ lder embedding between unit spheres of p-normed spaces ( 0 < p ≤ 1 ) under the quasi-H o ¨ lder or anti-H o ¨ lder type assumption. We generalize the main results in Xiao and Lu (Ann Funct Anal 13(2):Paper No. 25, 2022) from the r-isometric case to the wider case of quasi-H o ¨ lder mappings. Moreover, the method we adopted here is different; and the Hahn–Banach property, required as a condition in the main theorems of [30] has been removed in our results. [ABSTRACT FROM AUTHOR]

Let X, Y be real normed spaces and let ρ + ′ , ρ - ′ be norm derivatives. In this work, we solve a system of functional equations ρ + ′ (f (x) , f (y)) = g (x) ρ + ′ (x , y) , ρ - ′ (f (x) , f (y)) = g (x) ρ - ′ (x , y) , with unknown functions f : X → Y , g : X → R . Moreover, we give partial answer to open problem posed in 2010. [ABSTRACT FROM AUTHOR]

In this paper, we discover more proximinal closed convex subsets of unit spheres of some non-reflexive Banach spaces. Let B be a Banach space. For an element, u belongs to the unit sphere S B , and for a norm attaining functional μ on B, the sets Q(u) and J B (μ) are faces of S B. We consider B to be C 0 (X) , the space of all real-valued continuous functions vanishing at infinity on a locally compact Hausdorff space X, and L 1 (K , μ) , the space of all μ -measurable functions on a compact Hausdorff space K. In each space, we prove that the face Q(u) is strongly proximinal in B. For B = A (K) , the space of all affine continuous functions on a Choquet simplex K, we prove sufficient conditions for the faces Q(u) and J B (μ) to be proximinal in B. If Y is the space of all self-adjoint compact operators on l 2 , then Y is a closed subspace of K (l 2) , the space of all compact operators on l 2 . We characterize the faces Q(u) and J Y (μ) of Y and discuss their proximinality properties. [ABSTRACT FROM AUTHOR]

Let f : c 0 + ∪ - c 0 + → Y be a standard ε -isometry, where c 0 + is the positive cone of c 0 and Y is a Banach space. In this paper, we study stability properties of f and its symmetrization Θ = f (·) - f (- ·) 2 . We show that every standard isometry f is always stable. If f is a standard ε -isometry for ε > 0 , then f is stable for every separable space Y, i.e. there exists a bounded linear operator T : Y → c 0 with ‖ T ‖ ≤ 2 such that ‖ T f (x) - x ‖ ≤ 6 ε , for all x ∈ c 0 + ∪ - c 0 +. If f (c 0 + ∪ - c 0 +) contains a reproducing wedge of Y, then there exists a linear surjective isometry U : c 0 → Y such that f - U is uniformly bounded by 2 ε on c 0 + ∪ - c 0 + . We also obtain a series of new sharp stability results for Θ . [ABSTRACT FROM AUTHOR]

The real interpolation spaces between L p (R n) and H ˙ t , p (R n) (resp. H t , p (R n) ), t > 0 , are characterized in terms of fractional moduli of smoothness, and the underlying seminorms are shown to be "the correct" fractional generalization of the classical Gagliardo seminorms. This is confirmed by the fact that, using the new spaces combined with interpolation and extrapolation methods, we are able to extend the Bourgain–Brezis–Mironescu–Maz'ya–Shaposhnikova limit formulae, as well as the Bourgain–Brezis–Mironescu convergence theorem, to fractional Sobolev spaces. On the other hand, we disprove a conjecture of Brazke et al. (Bourgain–Brezis–Mironescu convergence via Triebel–Lizorkin spaces. https://arxiv.org/abs/2109.04159) suggesting fractional convergence results given in terms of classical Gagliardo seminorms. We also solve a problem proposed in Brazke et al. (Bourgain–Brezis–Mironescu convergence via Triebel–Lizorkin spaces. https://arxiv.org/abs/2109.04159) concerning sharp forms of the fractional Sobolev embedding. [ABSTRACT FROM AUTHOR]

In this paper, we examine the validity of bicomplex versions of some crucial inequalities with respect to the hyperbolic-valued norm | ⋅ | 한 and we discuss some topological and geometric concepts such as completeness, convexity, strict convexity and uniform convexity in the bicomplex setting with respect to the hyperbolic-valued norm ∥ ⋅ ∥ 픻 , ⋅ by defining the concept of 픻 -normed Banach bicomplex A-module and constructing 픻 -normed Banach bicomplex 픹 ⁢ ℂ -modules l p 한 ⁢ (픹 ⁢ ℂ) . [ABSTRACT FROM AUTHOR]

Let K be a compact Hausdorff perfectly normal space and T be a compact Hausdorff space, C + (K) = { f ∈ C (K) : f (k) ≥ 0 for all k ∈ K } be the positive cone of C(K). In this paper, we show that if F : C + (K) → C + (T) is a phase-isometry, that is { ‖ F (f) + F (g) ‖ , ‖ F (f) - F (g) ‖ } = { ‖ f + g ‖ , ‖ f - g ‖ } , ∀ f , g ∈ C + (K) , then there exists a nonempty closed subset S ⊂ T , such that F (·) | S : C + (K) → C + (S) (restriction of F (·) to S) is an additive isometry (the restriction of a linear isometry between C(K) and C(S)). Moreover, if F is almost surjective, then K and T are homeomorphic and F is the restriction of a surjective linear isometry between C(K) and C(T) induced by the homeomorphism. [ABSTRACT FROM AUTHOR]

In this paper, we present results concerning orthogonality in Hilbert C ∗ -modules. Moreover, for a C ∗ -algebra A , we prove theorems concerning the multi- A -linearity and its preservation by A -linear operators. New version of solution of the orthogonality equation on Hilbert C ∗ -modules and mappings preserving orthogonality are also investigated. [ABSTRACT FROM AUTHOR]

In this paper, we study two properties viz., property-U and property-SU of a subspace Y of a Banach space X, which correspond to the uniqueness of the Hahn–Banach extension of each linear functional in Y ∗ and when this association forms a linear operator of norm-1 from Y ∗ to X ∗ . It is proved that, under certain geometric assumptions on X, Y, Z, these properties are stable with respect to the injective tensor product; Y has property-U (SU) in Z if and only if X ⊗ ε ∨ Y has property-U (SU) in X ⊗ ε ∨ Z. We prove that when X ∗ has the Radon–Nikodým Property for 1 < p < ∞, Lp(μ, Y) has property-U (property-SU) in Lp(μ, X) if and only if Y is so in X. We show that if Z⊆ Y⊆ X and Y has property-U (SU) in X then Y/Z has property-U (SU) in X/Z. On the other hand, Y has property-SU in X if Y/Z has property-SU in X/Z and Z (⊆ Y) is an M-ideal in X. This partly solves the 3-space problem for property-SU. We characterize all hyperplanes in c0 which have property-SU. We derive necessary and sufficient conditions for all finite codimensional proximinal subspaces of c0 which have property-U (SU). [ABSTRACT FROM AUTHOR]

In this paper, we mainly study applications of the calmness moduli for multifunctions to error bounds of several non-convex systems. Based on the work given in Shen et al. [Calmness and the Abadie CQ for multifunctions and linear regularity for a collection of closed sets. SIAM J Optim. 2019;29(3):2291–2319], we use results on the calmness modulus of the multifunction therein to study error bounds of differentiable inclusions, weak sharp minima of a lower semicontinuous function and linear regularity of finitely many closed subsets. Several primal equivalent conditions for these regularity properties of the corresponding non-convex systems are provided with some mild assumptions. [ABSTRACT FROM AUTHOR]

In this paper we initiate the study of tropical Voronoi diagrams. We start out with investigating bisectors of finitely many points with respect to arbitrary polyhedral norms. For this more general scenario we show that bisectors of three points are homeomorphic to a non-empty open subset of Euclidean space, provided that certain degenerate cases are excluded. Specializing our results to tropical bisectors then yields structural results and algorithms for tropical Voronoi diagrams. [ABSTRACT FROM AUTHOR]

Recently, the authors interested some new problems on multiplier spaces of Lorentz' almost convergence and fλ-convergence as a generalization of almost convergence. fλ-convergence is firstly introduced by Karakuş and Başar, and used for some new characterizations of completeness and barrelledness of the spaces through weakly unconditionally Cauchy series in a normed space X and its continuous dual X*. In the present paper, we deal with fλ-convergence to have some inclusion relations between the vector valued spaces obtained from this type convergence and corresponding classical sequence spaces, and to give new characterizations of some classical properties like completeness, reflexivity, Schur property and Grothendieck property of normed spaces. By the way, we give a characterization of finite-dimensional normed spaces. [ABSTRACT FROM AUTHOR]

Take a point ξ on the boundary of a convex body F in R n , near which the boundary is given by an implicit equation. We present some notes on the formula, proposed in Pereira [A directional curvature formula for convex bodies in R n . J Math Anal Appl. 2022;506(1):125656.], for calculating the curvature of F at ξ in the direction of its any tangent vector. Namely, we see that our formula is equivalent to the existing one for the curvature of a certain curve given by the intersection of n−1 implicit equations, but it is easier to apply. Furthermore, we show that when the directional curvature of F is positive, there is the directional derivative of the Minkowski functional of the polar set F o , and we propose a formula to calculate it. [ABSTRACT FROM AUTHOR]

In this paper, we study a modified implicit rule for finding a solution of split common fixed point problem of a Bregman quasi-nonexpansive mapping in Banach spaces. We propose a new iterative algorithm and prove the strong convergence theorem under appropriate conditions. As an applica- tion, the results are applied to solving the zero problem and the equilibrium problem. [ABSTRACT FROM AUTHOR]

The skewness of Banach spaces was introduced by Fitzpatrick–Reznick. In this paper, we compute the skewness s (ℓ p - ℓ 1) of Day–James spaces ℓ p - ℓ 1 , where 1 < p < ∞ . This gives that the inequality s (X) ≤ 2 ρ X (1) is strict for such space X, where ρ X is the modulus of smoothness of X. [ABSTRACT FROM AUTHOR]

A wide new class of subsets of a Banach space X named coarse p-limited sets ( 1 ≤ p < ∞ ) is introduced by considering weak* p-summable sequences in X ′ instead of weak* null sequences. We study its basic properties and compare it with the class of compact and weakly compact sets. Results concerning the relationship of coarse p-limited sets with operators are obtained. [ABSTRACT FROM AUTHOR]