scholarly journals SYMMETRIC FINITE REPRESENTABILITY OF ℓp IN ORLICZ SPACES

2021 ◽  
Vol 26 (4) ◽  
pp. 15-24
Author(s):  
S. V. Astashkin
Keyword(s):  
Banach Space ◽  
Orlicz Space ◽  
Orlicz Spaces ◽  
Large Dimension ◽  
Complete Proof ◽  
Finite Dimensional ◽  
Finite Representability

It is well known that a Banach space need not contain any subspace isomorphic to a space ℓp (1 6 p ) or c0 (it was shown by Tsirelson in 1974). At the same time, by the famous Krivines theorem, every Banach space X always contains at least one of these spaces locally, i.e., there exist finite-dimensional subspaces of X of arbitrarily large dimension n which are isomorphic (uniformly) to ℓnp for some 1 6 p or cn0 . In thiscase one says that ℓp (resp. c0) is finitely representable in X. The main purpose of this paper is to give a characterization (with a complete proof) of the set of p such that ℓp is symmetrically finitely representable in a separable Orlicz space.

A note on symmetric basic sequences in Lp(Lq)

1992 ◽  
Vol 112 (1) ◽  
pp. 183-194
Author(s):  
Yves Raynaud
Keyword(s):  
Orlicz Space ◽  
Orlicz Spaces ◽  
Random Variables ◽  
Invariant Function ◽  
Function Spaces ◽  
Symmetric Basis ◽  
Finite Dimensional ◽  
Rearrangement Invariant Function Spaces ◽  
Independent Identically Distributed

Subspaces of Lp spanned by symmetric independent identically distributed random variables were identified as Orlicz spaces by Bretagnolle and Dacunha-Castelle[1], who showed that, conversely, in the case p ≤ 2, every p-convex, 2-concave Orlicz space is isomorphic to a subspace of Lp. This was extended by Dacunha-Castelle [3] to subspaces of Lp with symmetric basis, which appear as ‘p-means’ of Orlicz spaces (see [9] for the corresponding finite-dimensional result, and [12] for the case of rearrangement invariant function spaces). On the contrary the only subspaces with symmetric basis of Lp for p ≥ 2 are lp and l2 (if one does not care about isomorphy constants).

scholarly journals Regular Banach space net and abstract-valued Orlicz space of range-varying type

2019 ◽  
Vol 17 (1) ◽  
pp. 1680-1702
Author(s):  
Qinghua Zhang ◽  
Yueping Zhu
Keyword(s):  
Banach Space ◽  
Orlicz Space ◽  
Orlicz Spaces ◽  
Real Interpolation ◽  
Interpolation Spaces ◽  
Type Ii ◽  
The Real ◽  
Geometrical Properties

Abstract This paper investigates the abstract-valued Orlicz space of range-varying type. We firstly give the notions and examples of partially continuous modular net and regular Banach space net of type (II), then deal with the definitions, constructions, and geometrical properties of the range-varying Orlicz spaces, including representation of the dual $\begin{array}{} L_{+}^{\varphi} \end{array}$(I, Xθ(⋅))*, and reflexivity of Lφ(I, Xθ(⋅)), under some reasonable conditions. As an application, we finally make another approach to the real interpolation spaces constructed by a generalized Φ-function.

The wigner property for CL-spaces and finite-dimensional polyhedral banach spaces

2021 ◽  
pp. 1-17
Author(s):  
Dongni Tan ◽  
Xujian Huang
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Phase Function ◽  
Linear Isometry ◽  
Basic Properties ◽  
Finite Dimensional ◽  
Image Position

Abstract We say that a map $f$ from a Banach space $X$ to another Banach space $Y$ is a phase-isometry if the equality \[ \{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} \] holds for all $x,\,y\in X$ . A Banach space $X$ is said to have the Wigner property if for any Banach space $Y$ and every surjective phase-isometry $f : X\rightarrow Y$ , there exists a phase function $\varepsilon : X \rightarrow \{-1,\,1\}$ such that $\varepsilon \cdot f$ is a linear isometry. We present some basic properties of phase-isometries between two real Banach spaces. These enable us to show that all finite-dimensional polyhedral Banach spaces and CL-spaces possess the Wigner property.

scholarly journals Best Simultaneous Approximation in Orlicz Spaces

2007 ◽  
Vol 2007 ◽  
pp. 1-7 ◽  
Author(s):  
M. Khandaqji ◽  
Sh. Al-Sharif
Keyword(s):  
Banach Space ◽  
Simultaneous Approximation ◽  
Closed Subspace ◽  
Orlicz Spaces ◽  
Unit Interval ◽  
Luxemburg Norm ◽  
Integrable Functions ◽  
Best Simultaneous Approximation ◽  
Distance Formula

LetXbe a Banach space and letLΦ(I,X)denote the space of OrliczX-valued integrable functions on the unit intervalIequipped with the Luxemburg norm. In this paper, we present a distance formula dist(f1,f2,LΦ(I,G))Φ, whereGis a closed subspace ofX, andf1,f2∈LΦ(I,X). Moreover, some related results concerning best simultaneous approximation inLΦ(I,X)are presented.

scholarly journals On the convergence of greedy algorithms for initial segments of the Haar basis

2010 ◽  
Vol 148 (3) ◽  
pp. 519-529 ◽  
Author(s):  
S. J. DILWORTH ◽  
E. ODELL ◽  
TH. SCHLUMPRECHT ◽  
ANDRÁS ZSÁK
Keyword(s):  
Banach Space ◽  
Greedy Algorithm ◽  
General Result ◽  
Initial Segment ◽  
Greedy Algorithms ◽  
Dimensional Banach Space ◽  
Haar Basis ◽  
Finite Dimensional ◽  
Finite Dimensional Banach Space ◽  
Number Of Iterations

AbstractWe consider the X-Greedy Algorithm and the Dual Greedy Algorithm in a finite-dimensional Banach space with a strictly monotone basis as the dictionary. We show that when the dictionary is an initial segment of the Haar basis in Lp[0, 1] (1 < p < ∞) then the algorithms terminate after finitely many iterations and that the number of iterations is bounded by a function of the length of the initial segment. We also prove a more general result for a class of strictly monotone bases.

scholarly journals Some spectral formulas for functions generated by differential and integral operators in Orlicz spaces

2021 ◽  
Vol 13 (2) ◽  
pp. 326-339
Author(s):  
H.H. Bang ◽  
V.N. Huy
Keyword(s):  
Fourier Transform ◽  
Orlicz Space ◽  
Differential Operators ◽  
Orlicz Spaces ◽  
Integral Operators ◽  
Young Function ◽  
The Fourier Transform

In this paper, we investigate the behavior of the sequence of $L^\Phi$-norm of functions, which are generated by differential and integral operators through their spectra (the support of the Fourier transform of a function $f$ is called its spectrum and denoted by sp$(f)$). With $Q$ being a polynomial, we introduce the notion of $Q$-primitives, which will return to the notion of primitives if ${Q}(x)= x$, and study the behavior of the sequence of norm of $Q$-primitives of functions in Orlicz space $L^\Phi(\mathbb R^n)$. We have the following main result: let $\Phi $ be an arbitrary Young function, ${Q}({\bf x} )$ be a polynomial and $(\mathcal{Q}^mf)_{m=0}^\infty \subset L^\Phi(\mathbb R^n)$ satisfies $\mathcal{Q}^0f=f, {Q}(D)\mathcal{Q}^{m+1}f=\mathcal{Q}^mf$ for $m\in\mathbb{Z}_+$. Assume that sp$(f)$ is compact and $sp(\mathcal{Q}^{m}f)= sp(f)$ for all $m\in \mathbb{Z}_+.$ Then $$ \lim\limits_{m\to \infty } \|\mathcal{Q}^m f\|_{\Phi}^{1/m}= \sup\limits_{{\bf x} \in sp(f)} \bigl|1/ {Q}({\bf x}) \bigl|. $$ The corresponding results for functions generated by differential operators and integral operators are also given.

scholarly journals Geometric Invariants of Surjective Isometries between Unit Spheres

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2346
Author(s):  
Almudena Campos-Jiménez ◽  
Francisco Javier García-Pacheco
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Unit Sphere ◽  
Geometric Property ◽  
The Other ◽  
Geometric Invariants ◽  
Finite Dimensional ◽  
Surjective Isometry ◽  
Finite Dimensional Case

In this paper we provide new geometric invariants of surjective isometries between unit spheres of Banach spaces. Let X,Y be Banach spaces and let T:SX→SY be a surjective isometry. The most relevant geometric invariants under surjective isometries such as T are known to be the starlike sets, the maximal faces of the unit ball, and the antipodal points (in the finite-dimensional case). Here, new geometric invariants are found, such as almost flat sets, flat sets, starlike compatible sets, and starlike generated sets. Also, in this work, it is proved that if F is a maximal face of the unit ball containing inner points, then T(−F)=−T(F). We also show that if [x,y] is a non-trivial segment contained in the unit sphere such that T([x,y]) is convex, then T is affine on [x,y]. As a consequence, T is affine on every segment that is a maximal face. On the other hand, we introduce a new geometric property called property P, which states that every face of the unit ball is the intersection of all maximal faces containing it. This property has turned out to be, in a implicit way, a very useful tool to show that many Banach spaces enjoy the Mazur-Ulam property. Following this line, in this manuscript it is proved that every reflexive or separable Banach space with dimension greater than or equal to 2 can be equivalently renormed to fail property P.

Orlicz Spaces Without Strongly Extreme Points and Without H-Points

1993 ◽  
Vol 36 (2) ◽  
pp. 173-177 ◽  
Author(s):  
Henryk Hudzik
Keyword(s):  
Extreme Point ◽  
Orlicz Space ◽  
Unit Sphere ◽  
Orlicz Function ◽  
Orlicz Spaces ◽  
Extreme Points ◽  
Luxemburg Norm ◽  
Nonatomic Measure

AbstractW. Kurc [5] has proved that in the unit sphere of Orlicz space LΦ(μ) generated by an Orlicz function Φ satisfying the suitable Δ2-condition and equipped with the Luxemburg norm every extreme point is strongly extreme. In this paper it is proved in the case of a nonatomic measure μ that the unit sphere of the Orlicz space LΦ(μ) generated by an Orlicz function Φ which does not satisfy the suitable Δ2-condition and equipped with the Luxemburg norm has no strongly extreme point and no H-point.

scholarly journals Strongly Extreme Points and Middle Point Locally Uniformly Convex in Orlicz Spaces Equipped with s-Norm

2019 ◽  
Vol 2019 ◽  
pp. 1-7
Author(s):  
Yunan Cui ◽  
Yujia Zhan
Keyword(s):  
Banach Spaces ◽  
Orlicz Space ◽  
Necessary Conditions ◽  
Orlicz Function ◽  
Orlicz Spaces ◽  
Extreme Points ◽  
Uniformly Convex ◽  
Middle Point ◽  
Sufficient And Necessary Conditions

As is well known, the extreme points and strongly extreme points play important roles in Banach spaces. In this paper, the criterion for strongly extreme points in Orlicz spaces equipped with s-norm is given. We complete solved criterion-Orlicz space that generated by Orlicz function. And the sufficient and necessary conditions for middle point locally uniformly convex in Orlicz spaces equipped with s-norm are obtained.

scholarly journals Primal Lower Nice Functions in Reflexive Smooth Banach Spaces

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2066
Author(s):  
Messaoud Bounkhel ◽  
Mostafa Bachar
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Space Setting ◽  
Nonlinear Anal ◽  
Finite Dimensional ◽  
In Trans ◽  
First Time ◽  
The Relationship ◽  
Primal Lower Nice Functions

In the present work, we extend, to the setting of reflexive smooth Banach spaces, the class of primal lower nice functions, which was proposed, for the first time, in finite dimensional spaces in [Nonlinear Anal. 1991, 17, 385–398] and enlarged to Hilbert spaces in [Trans. Am. Math. Soc. 1995, 347, 1269–1294]. Our principal target is to extend some existing characterisations of this class to our Banach space setting and to study the relationship between this concept and the generalised V-prox-regularity of the epigraphs in the sense proposed recently by the authors in [J. Math. Anal. Appl. 2019, 475, 699–29].

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