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Transitivity of proximinality and norm attaining functionals

Darapaneni Narayana, T. S. S. R. K. Rao (2006)

Colloquium Mathematicae

We study the question of when the set of norm attaining functionals on a Banach space is a linear space. We show that this property is preserved by factor reflexive proximinal subspaces in R ( 1 ) ˜ spaces and generally by taking quotients by proximinal subspaces. We show, for (ℓ₂) and c₀-direct sums of families of reflexive spaces, the transitivity of proximinality for factor reflexive subspaces. We also investigate the linear structure of the set of norm attaining functionals on hyperplanes of c₀ and...

Triebel-Lizorkin spaces with non-doubling measures

Yongsheng Han, Dachun Yang (2004)

Studia Mathematica

Suppose that μ is a Radon measure on d , which may be non-doubling. The only condition assumed on μ is a growth condition, namely, there is a constant C₀ > 0 such that for all x ∈ supp(μ) and r > 0, μ(B(x,r)) ≤ C₀rⁿ, where 0 < n ≤ d. The authors provide a theory of Triebel-Lizorkin spaces p q s ( μ ) for 1 < p < ∞, 1 ≤ q ≤ ∞ and |s| < θ, where θ > 0 is a real number which depends on the non-doubling measure μ, C₀, n and d. The method does not use the vector-valued maximal function inequality...

Two geometric constants for operators acting on a separable Banach space.

E. Martín Peinador, E. Induráin, A. Plans Sanz de Bremond, A. A. Rodes Usan (1988)

Revista Matemática de la Universidad Complutense de Madrid

The main result of this paper is the following: A separable Banach space X is reflexive if and only if the infimum of the Gelfand numbers of any bounded linear operator defined on X can be computed by means of just one sequence on nested, closed, finite codimensional subspaces with null intersection.

Two mappings related to semi-inner products and their applications in geometry of normed linear spaces

Sever Silvestru Dragomir, Jaromír J. Koliha (2000)

Applications of Mathematics

In this paper we introduce two mappings associated with the lower and upper semi-inner product ( · , · ) i and ( · , · ) s and with semi-inner products [ · , · ] (in the sense of Lumer) which generate the norm of a real normed linear space, and study properties of monotonicity and boundedness of these mappings. We give a refinement of the Schwarz inequality, applications to the Birkhoff orthogonality, to smoothness of normed linear spaces as well as to the characterization of best approximants.

Type and cotype of operator spaces

Hun Hee Lee (2008)

Studia Mathematica

We consider two operator space versions of type and cotype, namely S p -type, S q -cotype and type (p,H), cotype (q,H) for a homogeneous Hilbertian operator space H and 1 ≤ p ≤ 2 ≤ q ≤ ∞, generalizing “OH-cotype 2” of G. Pisier. We compute type and cotype of some Hilbertian operator spaces and L p spaces, and we investigate the relationship between a homogeneous Hilbertian space H and operator spaces with cotype (2,H). As applications we consider operator space versions of generalized little Grothendieck’s...

Types of tightness in spaces with unconditional basis

Antonis Manoussakis, Anna Pelczar-Barwacz (2014)

Studia Mathematica

We present a reflexive Banach space with an unconditional basis which is quasi-minimal and tight by range, i.e. of type (4) in Ferenczi-Rosendal's list related to Gowers' classification program of Banach spaces, but in contrast to the recently constructed space of type (4), our space is also tight with constants, thus essentially extending the list of known examples in Gowers' program. The space is defined on the basis of a boundedly modified mixed Tsirelson space with the use of a special coding...

Types on stable Banach spaces

José Iovino (1998)

Fundamenta Mathematicae

 We prove a geometric characterization of Banach space stability. We show that a Banach space X is stable if and only if the following condition holds. Whenever X ^ is an ultrapower of X and B is a ball in X ^ , the intersection B ∩ X can be uniformly approximated by finite unions and intersections of balls in X; furthermore, the radius of these balls can be taken arbitrarily close to the radius of B, and the norm of their centers arbitrarily close to the norm of the center of B.  The preceding condition...

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