Porosity, Γ‎N- and Γ‎-Null Sets

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Modulus Of Smoothness ◽  
Separable Space ◽  
Bump Function ◽  
Asplund Spaces ◽  
Porous Sets

This chapter introduces the notion of porosity “at infinity” (formally defined as porosity with respect to a family of subspaces) and discusses the main result, which shows that sets porous with respect to a family of subspaces are Γ‎ₙ-null provided X admits a continuous bump function whose modulus of smoothness (in the direction of this family) is controlled by tⁿ logⁿ⁻¹ (1/t). The first of these results characterizes Asplund spaces: it is shown that a separable space has separable dual if and only if all its porous sets are Γ‎₁-null. The chapter first describes porous and σ‎-porous sets as well as a criterion of Γ‎ₙ-nullness of porous sets. It then considers the link between directional porosity and Γ‎ₙ-nullness. Finally, it tackles the question in which spaces, and for what values of n, porous sets are Γ‎ₙ-null.

scholarly journals A note on bump functions that locally depend on finitely many coordinates

1997 ◽  
Vol 56 (3) ◽  
pp. 447-451 ◽  
Author(s):  
M. Fabian ◽  
V. Zizler
Keyword(s):  
Banach Space ◽  
Linear Span ◽  
Bump Function ◽  
Closed Linear Span ◽  
Asplund Spaces

We show that if a continuous bump function on a Banach space X locally depends on finitely many elements of a set F in X*, then the norm closed linear span of F equals to X*. Some corollaries for Markuševič bases and Asplund spaces are derived.

Fr échet Differentiability of Real-Valued Functions

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Variational Principles ◽  
Mean Value ◽  
Lipschitz Functions ◽  
Monotone Functions ◽  
Porous Set ◽  
Asplund Spaces ◽  
Fréchet Differentiability ◽  
The Mean ◽  
Frechet Differentiability ◽  
Porous Sets

This chapter shows that cone-monotone functions on Asplund spaces have points of Fréchet differentiability and that the appropriate version of the mean value estimates holds. It also proves that the corresponding point of Fréchet differentiability may be found outside any given σ‎-porous set. This new result considerably strengthens known Fréchet differentiability results for real-valued Lipschitz functions on such spaces. The avoidance of σ‎-porous sets is new even in the Lipschitz case. The chapter first discusses the use of variational principles to prove Fréchet differentiability before analyzing a one-dimensional mean value problem in relation to Lipschitz functions. It shows that results on existence of points of Fréchet differentiability may be generalized to derivatives other than the Fréchet derivative.

THE STRUCTURE OF THE σ-IDEAL OF σ-POROUS SETS

1999 ◽  
Vol 25 (1) ◽  
pp. 87
Author(s):  
Zelený
Keyword(s):  
Porous Sets

Projections on Tree-Like banach Spaces

1985 ◽  
Vol 37 (5) ◽  
pp. 908-920
Author(s):  
A. D. Andrew
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Bounded Linear Operator ◽  
Unconditional Basis ◽  
Reflexive Banach Space ◽  
Separable Space ◽  
Bounded Linear ◽  
Tree Space

1. In this paper, we investigate the ranges of projections on certain Banach spaces of functions defined on a diadic tree. The notion of a “tree-like” Banach space is due to James 4], who used it to construct the separable space JT which has nonseparable dual and yet does not contain l1. This idea has proved useful. In [3], Hagler constructed a hereditarily c0 tree space, HT, and Schechtman [6] constructed, for each 1 ≦ p ≦ ∞, a reflexive Banach space, STp with a 1-unconditional basis which does not contain lp yet is uniformly isomorphic to for each n.In [1] we showed that if U is a bounded linear operator on JT, then there exists a subspace W ⊂ JT, isomorphic to JT such that either U or (1 — U) acts as an isomorphism on W and UW or (1 — U)W is complemented in JT. In this paper, we establish this result for the Hagler and Schechtman tree spaces.

scholarly journals Quantitative Estimates for Nonlinear Sampling Kantorovich Operators

2021 ◽  
Vol 76 (2) ◽  
Author(s):  
Nursel Çetin ◽  
Danilo Costarelli ◽  
Gianluca Vinti
Keyword(s):  
Orlicz Spaces ◽  
General Setting ◽  
Modulus Of Smoothness ◽  
Direct Approach ◽  
Order Of Approximation ◽  
Lipschitz Classes ◽  
General Frame ◽  
Quantitative Estimates ◽  
Nonlinear Sampling ◽  
Kantorovich Operators

AbstractIn this paper, we establish quantitative estimates for nonlinear sampling Kantorovich operators in terms of the modulus of smoothness in the setting of Orlicz spaces. This general frame allows us to directly deduce some quantitative estimates of approximation in $$L^{p}$$ L p -spaces, $$1\le p<\infty $$ 1 ≤ p < ∞ , and in other well-known instances of Orlicz spaces, such as the Zygmung and the exponential spaces. Further, the qualitative order of approximation has been obtained assuming f in suitable Lipschitz classes. The above estimates achieved in the general setting of Orlicz spaces, have been also improved in the $$L^p$$ L p -case, using a direct approach suitable to this context. At the end, we consider the particular cases of the nonlinear sampling Kantorovich operators constructed by using some special kernels.

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