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Narrow operators and rich subspaces of Banach spaces with the Daugavet property

Vladimir M. Kadets, Roman V. Shvidkoy, Dirk Werner (2001)

Studia Mathematica

Let X be a Banach space. We introduce a formal approach which seems to be useful in the study of those properties of operators on X which depend only on the norms of the images of elements. This approach is applied to the Daugavet equation for norms of operators; in particular we develop a general theory of narrow operators and rich subspaces of spaces X with the Daugavet property previously studied in the context of the classical spaces C(K) and L₁(μ).

Near smoothness of Banach spaces.

Józef Banas, Kishin Sadarangani (1995)

Collectanea Mathematica

The aim of this paper is to discuss the concept of near smoothness in some Banach sequence spaces.

Nearly smooth points and near smoothness in Orlicz spaces

Ji Donghai, Yanming Lü, Ting Fu Wang (1998)

Commentationes Mathematicae Universitatis Carolinae

Nearly smooth points and near smoothness in Orlicz spaces are characterized. It is worth to notice that in the nonatomic case smooth points and nearly smooth points are the same, but in the sequence case they differ.

Non-containment of l1 in projective tensor products of Banach spaces.

J. C. Díaz Alcaide (1990)

Revista Matemática de la Universidad Complutense de Madrid

Two properties on projective tensor products are introduced and briefly studied. We apply them to give sufficient conditions to assure the non-containment of l1 in a projective tensor product of Banach spaces.

Non-universal families of separable Banach spaces

Ondřej Kurka (2016)

Studia Mathematica

We prove that if 𝓒 is a family of separable Banach spaces which is analytic with respect to the Effros Borel structure and no X ∈ 𝓒 is isometrically universal for all separable Banach spaces, then there exists a separable Banach space with a monotone Schauder basis which is isometrically universal for 𝓒 but not for all separable Banach spaces. We also establish an analogous result for the class of strictly convex spaces.

Norm attaining multilinear forms and polynomials on preduals of Lorentz sequence spaces

M. Jimenéz Sevilla, Rafael Payá (1998)

Studia Mathematica

For each natural number N, we give an example of a Banach space X such that the set of norm attaining N-linear forms is dense in the space of all continuous N-linear forms on X, but there are continuous (N+1)-linear forms on X which cannot be approximated by norm attaining (N+1)-linear forms. Actually,X is the canonical predual of a suitable Lorentz sequence space. We also get the analogous result for homogeneous polynomials.

Norm attaining operators versus bilinear forms.

Rafael Payá (1997)

Extracta Mathematicae

The well known Bishop-Phelps Theorem asserts that the set of norm attaining linear forms on a Banach space is dense in the dual space [3]. This note is an outline of recent results by Y. S. Choi [5] and C. Finet and the author [7], which clarify the relation between two different ways of extending this theorem.

Norm continuity of weakly quasi-continuous mappings

Alireza Kamel Mirmostafaee (2011)

Colloquium Mathematicae

Let be the class of Banach spaces X for which every weakly quasi-continuous mapping f: A → X defined on an α-favorable space A is norm continuous at the points of a dense G δ subset of A. We will show that this class is stable under c₀-sums and ℓ p -sums of Banach spaces for 1 ≤ p < ∞.

Normal structure and weakly normal structure of Orlicz spaces

Shutao Chen, Yanzheng Duan (1991)

Commentationes Mathematicae Universitatis Carolinae

Every Orlicz space equipped with Orlicz norm has weak sum property, therefore, it has weakly normal structure and fixed point property. A criterion of sum property also of normal structure for such spaces is given as well, which shows that every Orlicz space has isonormal structure.

Normal structure of Lorentz-Orlicz spaces

Pei-Kee Lin, Huiying Sun (1997)

Annales Polonici Mathematici

Let ϕ: ℝ → ℝ₊ ∪ 0 be an even convex continuous function with ϕ(0) = 0 and ϕ(u) > 0 for all u > 0 and let w be a weight function. u₀ and v₀ are defined by u₀ = supu: ϕ is linear on (0,u), v₀=supv: w is constant on (0,v) (where sup∅ = 0). We prove the following theorem. Theorem. Suppose that Λ ϕ , w ( 0 , ∞ ) (respectively, Λ ϕ , w ( 0 , 1 ) ) is an order continuous Lorentz-Orlicz space. (1) Λ ϕ , w has normal structure if and only if u₀ = 0 (respectively, ∫ v ₀ ϕ ( u ₀ ) · w < 2 a n d u ₀ < ∞ ) . (2) Λ ϕ , w has weakly normal structure if and only if ∫ 0 v ₀ ϕ ( u ₀ ) · w < 2 .

Norm-attaining polynomials and differentiability

Juan Ferrera (2002)

Studia Mathematica

We give a polynomial version of Shmul'yan's Test, characterizing the polynomials that strongly attain their norm as those at which the norm is Fréchet differentiable. We also characterize the Gateaux differentiability of the norm. Finally we study those properties for some classical Banach spaces.

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