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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]

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]

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]

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]

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]

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]

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]

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]

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]

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 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 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]

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]

In this paper, we show that every function in an anisotropic fractional Sobolev space defined on a regular domain extends to a function defined on the whole R n with the same Sobolev indices. As an application, we get that such functions are Hölder continuous and we give an explicit estimate for the continuity exponent. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of M 2 -strict projections in M 2 (M 0) where M 0 is an abelian von Neumann algebra and prove that an absolutely compatible pair of strict elements in a von Neumann algebra M is unitarily equivalent to the pair of elements (1 - x 0) ⊗ I 2 P 0 + (x 0 ⊗ I 2) P and (1 - x 0) ⊗ I 2 P 0 + (x 0 ⊗ I 2) P ′ in M 2 (M 0) where M 0 is an abelian von Neumann algebra, x 0 is a strict element of M 0 + , P 0 = 0 0 0 1 ∈ M 2 (M 0) and P is an M 2 -strict projection in M 2 (M 0) . We also discuss the geometric form of this representation when M = M 2 . [ABSTRACT FROM AUTHOR]

A nonlinear characterization of type I factors among von Neumann algebras based on strong Birkhoff–James orthogonality is given. As an application, it is shown that if a von Neumann algebra is isomorphic to a type I factor with respect to the structure of strong Birkhoff–James orthogonality, then they are ∗ -isomorphic as von Neumann algebras. [ABSTRACT FROM AUTHOR]

In this paper, the extension of isometries is considered for some operators between the unit spheres of two real p-normed spaces ( 0 < p ≤ 1 ) under some nonexpansivity and expansivity assumptions. In the light of the characteristics of p-normed spaces, the notion of isometry is generalized to the case of r-isometries. By virtue of the technique of dual spaces, it is proved that such operators can be extended from the unit sphere to the whole space. [ABSTRACT FROM AUTHOR]

The design of optimal Magnetic Resonance Imaging (MRI) coils is modeled as a minimum-norm problem (MNP), that is, as an optimization problem of the form min x ∈ R ∥ x ∥ , where R is a closed and convex subset of a normed space X. This manuscript is aimed at revisiting MNPs from the perspective of Functional Analysis, Operator Theory, and Banach Space Geometry in order to provide an analytic solution to the following MRI problem: min ψ ∈ R ∥ ψ ∥ 2 , where R : = { ψ ∈ R n : ∥ A ψ − b ∥ ∞ ∥ b ∥ ∞ ≤ D } , with A ∈ M m × n (R) , D > 0 , and b ∈ R m ∖ { 0 } . [ABSTRACT FROM AUTHOR]

Let X, Y be two w ∗ -almost smooth Banach spaces, C (B (X ∗) , w ∗) be the Banach space of all continuous real-valued functions on B (X ∗) endowed with the supremum norm and C + (B (X ∗) , w ∗) be the positive cone of C (B (X ∗) , w ∗) . In this paper, we show that if F : C + (B (X ∗) , w ∗) → C + (B (Y ∗) , w ∗) is a standard almost surjective ε -isometry, then there exists a homeomorphism τ : B (X ∗) → B (Y ∗) in the w ∗ -topology such that for any x ∗ ∈ B (X ∗) , we have | ⟨ δ x ∗ , f ⟩ - ⟨ δ τ (x ∗) , F (f) ⟩ | ≤ 2 ε , for all f ∈ C + (B (X ∗) , w ∗). As its application, we show that if U : C (B (X ∗) , w ∗) → C (B (Y ∗) , w ∗) is the canonical linear surjective isometry induced by the homeomorphism γ = τ - 1 : B (Y ∗) → B (X ∗) in the w ∗ -topology, then ‖ F (f) - U (f) ‖ ≤ 2 ε , for all f ∈ C + (B (X ∗) , w ∗). [ABSTRACT FROM AUTHOR]

We study the norm derivatives in the context of Birkhoff–James orthogonality in real Banach spaces. As an application of this, we obtain a complete characterization of the left-symmetric points and the right-symmetric points in a real Banach space in terms of the norm derivatives. We obtain a complete characterization of strong Birkhoff–James orthogonality in ℓ 1 n and ℓ ∞ n spaces. We also obtain a complete characterization of the orthogonality relation defined by the norm derivatives in terms of some newly introduced variation of Birkhoff–James orthogonality. We further study Birkhoff–James orthogonality, approximate Birkhoff–James orthogonality, smoothness and norm attainment of bounded bilinear operators between Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study a Bishop–Phelps–Bollobás type property called the property L o , o of a pair of Banach spaces. Getting motivated by this, we introduce the notion of Approximate minimizing property (AMp) of a pair of Banach spaces and characterize finite dimensionality of Banach spaces with respect to this property. We further introduce the notion of approximate minimum norm attainment set of a bounded linear operator and characterize the AMp with the help of Hausdorff convergence of the sequence of approximate minimum norm attainment sets of bounded linear operators. We also investigate sufficient conditions for the holding of some weaker forms of the AMp for a pair of Banach spaces. Finally, we define and study uniform ε -approximation of a bounded linear operator in terms of its minimum norm. [ABSTRACT FROM AUTHOR]

In this paper, the necessary and sufficient conditions were given for Orlicz–Lorentz function space endowed with the Orlicz norm having uniformly normal structure and uniform non-squareness. [ABSTRACT FROM AUTHOR]

Sets in Banach spaces that are mapped into norm compact sets by operators T : X → ℓ p (called weakly p-Dunford Pettis sets), for 1 < p < ∞ , are studied in arbitrary Banach spaces X and in the space L 1 (μ , X) of Bochner integrable functions. Sufficient conditions for a subset of L 1 (μ , X) to be a weakly p-Dunford Pettis set are given. It is shown that if X ∗ ∈ C p , and K is a bounded and uniformly L p -integrable subset of L p (μ , X) , then K is a weakly p-DP set in L 1 (μ , X) . [ABSTRACT FROM AUTHOR]

We say that a map T : S X → S Y between the unit spheres of two real normed-spaces X and Y is a phase-isometry if it satisfies ‖ T (x) + T (y) ‖ , ‖ T (x) - T (y) ‖ = ‖ x + y ‖ , ‖ x - y ‖ for all x , y ∈ S X . In the present paper, we show that there is a phase function ε : S X → { - 1 , 1 } such that ε · T is an isometry which can be extended a real linear isometry on X whenever T is surjective, X = C (K) and Y is an arbitrary Banach space. Additionally, if T is a surjective phase-isometry between the unit spheres of C(K) and C (Ω) , where K and Ω are compact Hausdorff spaces, we prove that there are a homeomorphism φ : Ω → K and a continuous unimodular function h on Ω such that T (f) ∈ { h · f ∘ φ , - h · f ∘ φ } for all f ∈ S C (K) . This also can be seen as a Banach-Stone type representation for phase-isometries in C(K) spaces. [ABSTRACT FROM AUTHOR]

In this paper we characterize the Birkhoff–James orthogonality with respect to the numerical radius norm v (·) in C ∗ -algebras. More precisely, for two elements a, b in a C ∗ -algebra A , we show that a ⊥ B v b if and only if for each θ ∈ [ 0 , 2 π) , there exists a state φ θ on A such that | φ θ (a) | = v (a) and Re (e i θ φ θ (a) ¯ φ θ (b)) ≥ 0 . Moreover, we compute the numerical radius derivatives in A . In addition, we characterize when the numerical radius norm of the sum of two (or three) elements in A equals the sum of their numerical radius norms. [ABSTRACT FROM AUTHOR]

We introduce a concept of complete proximal normal structure and used to investigate the existence of a best proximity point for an arbitrary family of cyclic relatively nonexpansive mappings in the setting of strictly convex Banach spaces. We also prove that every bounded, closed and convex pair in uniformly convex Banach spaces as well as every compact and convex pairs in Banach spaces has complete proximal normal structure. Furthermore, we consider a class of cyclic relatively nonexpansive mappings in the sense of Suzuki and establish a new best proximity point theorem in the setting of Hilbert spaces. [ABSTRACT FROM AUTHOR]

In this paper, complete characterizations of left (or right) symmetric points for strong Birkhoff orthogonality in B (H , K) and K (H , K) are given, where H , K are complex Hilbert spaces and B (H , K) ( K (H , K) ) is the space of all bounded linear (compact) operators from H into K . [ABSTRACT FROM AUTHOR]

We study geometric properties of GL-spaces. We demonstrate that every finite-dimensional GL-space is polyhedral; that in dimension 2 there are only two, up to isometry, GL-spaces, namely the space whose unit sphere is a square (like ℓ ∞ 2 or ℓ 1 2 ) and the space whose unit sphere is an equilateral hexagon. Finally, we characterise the spaces E = (R n , ‖ · ‖ E) with absolute norm such that for every collection X 1 , ... , X n of GL-spaces their E-sum is a GL-space. [ABSTRACT FROM AUTHOR]

We present a complete characterization of the metric compactification of L p spaces for 1 ≤ p < ∞ . Each element of the metric compactification of L p is represented by a random measure on a certain Polish space. By way of illustration, we revisit the L p -mean ergodic theorem for 1 < p < ∞ , and Alspach's example of an isometry on a weakly compact convex subset of L 1 with no fixed points. [ABSTRACT FROM AUTHOR]

The paper deals with the achievements of introducing the notion of F-cone metric spaces over Fréchet algebra as a generalization of F-cone metric spaces over a Banach algebra, N p -cone metric spaces over a Banach algebra, and N b -cone metric spaces over a Banach algebra. First, we study some of its topological properties. Next, we define a generalized Lipschitz for such spaces. Also, we investigate some fixed points for mappings satisfying such conditions in the new framework. Subsequently, as an application of our results, we provide an example. Our work generalizes some well-known results in the literature. [ABSTRACT FROM AUTHOR]

In this present work, the topological structure of 2-normed spaces is studied from the viewpoint of topological vector spaces. A separating family of seminorms is induced from a given 2-norm, and the criteria for metrizability and normability of 2-normed spaces are investigated using this family. A sufficient condition for metrizability and a necessary and sufficient condition for normability of 2-normed spaces are deduced during this investigation. [ABSTRACT FROM AUTHOR]

We characterize the subspaces X of ℓ 1 satisfying Grothendieck's theorem in terms of extension of nonnegative quadratic forms q : X ⟶ R to the whole ℓ 1 . [ABSTRACT FROM AUTHOR]

Let W and Z be Banach spaces such that Z is separable and let R : W ⟶ Z be a (continuous, linear) operator. We study consequences of the adjoint operator R ⁎ having non-separable range. From our main technical result we obtain applications to the theory of basic sequences and the existence of universal operators for various classes of operators between Banach spaces. We also obtain an operator-theoretic characterisation of separable Banach spaces with non-separable dual. [ABSTRACT FROM AUTHOR]

In this article, the bi-Lipschitz embeddability of the sequence of countably branching diamond graphs ( D k ω ) k ∈ N is investigated. In particular it is shown that for every ε > 0 and k ∈ N , D k ω embeds bi-Lipschiztly with distortion at most 6 ( 1 + ε ) into any reflexive Banach space with an unconditional asymptotic structure that does not admit an equivalent asymptotically uniformly convex norm. On the other hand it is shown that the sequence ( D k ω ) k ∈ N does not admit an equi-bi-Lipschitz embedding into any Banach space that has an equivalent asymptotically midpoint uniformly convex norm. Combining these two results one obtains a metric characterization in terms of graph preclusion of the class of asymptotically uniformly convexifiable spaces, within the class of reflexive Banach spaces with an unconditional asymptotic structure. Applications to bi-Lipschitz embeddability into L p -spaces and to some problems in renorming theory are also discussed. [ABSTRACT FROM AUTHOR]

In a normed space we consider an approximate orthogonality relation related to the Birkhoff orthogonality. We give some properties of this relation as well as applications. In particular, we characterize the approximate orthogonality in the class of linear bounded operators on a Hilbert space. [ABSTRACT FROM AUTHOR]

We prove an explicit characterization of all injective subsets of the model space l ∞ ( I ) of all bounded real-valued functions defined on a non-empty set I , endowed with the supremum norm. Since the class of all injective metric spaces coincides with the one of all absolute 1-Lipschitz retracts, the present work yields a characterization of all the subsets of l ∞ ( I ) that are absolute 1-Lipschitz retracts. [ABSTRACT FROM AUTHOR]

We investigate the existence of solutions of the Kirchhoff elliptic equations with nonlinearity in $R^{N}$ . Using the ideas developed in Orlicz spaces and the technique of variation principle, we prove that there are at least three solutions in Orlicz-Sobolev spaces. [ABSTRACT FROM AUTHOR]

We show a new characterisation of the existence of smooth partitions of unity on a Banach space. This leads to a slight generalisation of some as well as a very easy recovery of most of the known results using a unified treatment. [ABSTRACT FROM AUTHOR]

We construct a Bourgain–Delbaen L ∞ -space X K u s with structure that is strongly heterogeneous: any bounded operator on X K u s is a compact perturbation of a multiple of the identity, whereas the space X K u s is saturated with unconditional basic sequences. [ABSTRACT FROM AUTHOR]

The paper deals with the achievement of introducing the notion of F-cone metric spaces over Banach algebra as a generalization of $N_{p}$ -cone metric space over Banach algebra and $N_{b}$ -cone metric space over Banach algebra and studying some of its topological properties. Also, here we define generalized Lipschitz and expansive maps for such spaces. Moreover, we investigate some fixed points for mappings satisfying such conditions in the new framework. Subsequently, as an application of our results, we provide an example. Our results generalize some well-known results in the literature. [ABSTRACT FROM AUTHOR]

We study the lattice structure of the family of weakly compact subsets of the unit ball B X of a separable Banach space X , equipped with the inclusion relation (this structure is denoted by K ( B X ) ) and also with the parametrized family of “almost inclusion” relations K ⊆ L + ε B X , where ε > 0 (this structure is denoted by AK ( B X ) ). Tukey equivalence between partially ordered sets and a suitable extension to deal with AK ( B X ) are used. Assuming the axiom of analytic determinacy, we prove that separable Banach spaces fall into four categories, namely: K ( B X ) is equivalent either to a singleton, or to ω ω , or to the family K ( Q ) of compact subsets of the rational numbers, or to the family [ c ] < ω of all finite subsets of the continuum. Also under the axiom of analytic determinacy, a similar classification of AK ( B X ) is obtained. For separable Banach spaces not containing ℓ 1 , we prove in ZFC that K ( B X ) ∼ AK ( B X ) are equivalent to either {0}, ω ω , K ( Q ) or [ c ] < ω . The lattice structure of the family of all weakly null subsequences of an unconditional basis is also studied. [ABSTRACT FROM AUTHOR]

Let ( M 1 , d 1 ) , ( M 2 , d 2 ) be metric spaces. A map f : M 1 → M 2 is said to be locally open at an x 1 ∈ M 1 , if for every ε > 0 one finds a δ > 0 such that B ( f ( x 1 ) , δ ) ⊂ f ( B ( x 1 , ε ) ) ; here B ( x , r ) stands for the closed ball with center x and radius r . We are particularly interested in the following special case: X , Y , Z are normed spaces, the spaces L ( X , Y ) , L ( Y , Z ) , L ( X , Z ) of linear continuous operators are provided with the operator norm, and the map under consideration is the bilinear map ( S , T ) ↦ S ∘ T (from L ( Y , Z ) × ( L ( X , Y ) to L ( X , Z ) ). For which pairs ( S 0 , T 0 ) ∈ L ( Y , Z ) × ( L ( X , Y ) is it locally open? The main result of the paper gives a complete characterization of pairs ( S , T ) at which this map is locally open in the case of finite-dimensional spaces X , Y , Z . [ABSTRACT FROM AUTHOR]

The Gurariĭ space is the unique separable Banach space G which is of almost universal disposition for finite-dimensional Banach spaces, which means that for every ε > 0 , for all finite-dimensional normed spaces E ⊆ F , for every isometric embedding e : E → G there exists an ε -isometric embedding f : F → G such that f ↾ E = e . We show that G N with a special sequence of semi-norms is of almost universal disposition for finite-dimensional graded Fréchet spaces. The construction relies heavily on the universal operator on the Gurariĭ space, recently constructed by Garbulińska-Węgrzyn and the third author. In addition, we consider a non-graded sequence of semi-norms on G N with which the space G N is of almost universal disposition for finite-dimensional Fréchet spaces with a fixed sequence of semi-norms. In both cases, this yields in particular that G N is universal in the class of all separable Fréchet spaces. [ABSTRACT FROM AUTHOR]

We study the general measures of non-compactness defined on subsets of a dual Banach space, their associated derivations and their ω -iterates. We introduce the notions of convexifiable and sublinear measure of non-compactness and investigate the properties of its associated fragment and slice derivations. We apply our results to the Kuratowski measure of non-compactness and to the study of the Szlenk index of a Banach space. As a consequence, we obtain that the Szlenk index and the convex Szlenk index of a separable Banach space are always equal. We also give, for any countable ordinal α , a characterization of the Banach spaces with Szlenk index bounded by ω α + 1 in terms of the existence of an equivalent renorming. This extends a result by Knaust, Odell and Schlumprecht on Banach spaces with Szlenk index equal to ω . [ABSTRACT FROM AUTHOR]

We investigate a correspondence between strict K -monotonicity, K -order continuity and the best dominated approximation problems with respect to the Hardy–Littlewood–Pólya relation ≺ . Namely, we study, in terms of an LKM point and a UKM point, a necessary condition for uniqueness of the best dominated approximation under the relation ≺ in a symmetric space E . Next, we characterize a relation between a point of K -order continuity and an existence of a best dominated approximant with respect to ≺ . Finally, we present a compete criteria, written in a notion of K -order continuity, under which a closed and K -bounded above subset of a symmetric space E is proximinal. [ABSTRACT FROM AUTHOR]