The superreflexivity property of a Banach space in terms of the closeness of its finite-dimensional subspaces to euclidean spaces

1978 ◽  
Vol 12 (2) ◽  
pp. 142-144 ◽  
Author(s):  
M. I. Kadets
Keyword(s):  
Banach Space ◽  
Euclidean Spaces ◽  
Finite Dimensional

THE PERSISTENCE OF SNAP-BACK REPELLER UNDER SMALL C1 PERTURBATIONS IN BANACH SPACES

2011 ◽  
Vol 21 (03) ◽  
pp. 703-710 ◽  
Author(s):  
YUANLONG CHEN ◽  
YU HUANG ◽  
LIANGLIANG LI
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Chaotic Behavior ◽  
Population Models ◽  
Euclidean Spaces ◽  
Finite Dimensional

In this paper, we consider the persistence of snap-back repellers under small C1 perturbations in Banach spaces. Let X be a Banach space and f be a C1-map from X into itself. We show that if f has a snap-back repeller, then any small C1 perturbations of f has a snap-back repeller and exhibits chaos in the sense of Devaney. The obtained results further extend the existing relevant results in finite-dimensional Euclidean spaces. As applications, we will discuss the chaotic behavior of two nonlocal population models.

scholarly journals Smooth Approximation of Lipschitz Maps and Their Subgradients

2022 ◽  
Vol 69 (1) ◽  
pp. 1-32
Author(s):  
Abbas Edalat
Keyword(s):  
Banach Space ◽  
Directional Derivative ◽  
Directional Derivatives ◽  
Euclidean Spaces ◽  
Smooth Approximation ◽  
Norm Topology ◽  
Lipschitz Maps ◽  
Finite Dimensional ◽  
Lipschitz Map ◽  
Locally Lipschitz

We derive new representations for the generalised Jacobian of a locally Lipschitz map between finite dimensional real Euclidean spaces as the lower limit (i.e., limit inferior) of the classical derivative of the map where it exists. The new representations lead to significantly shorter proofs for the basic properties of the subgradient and the generalised Jacobian including the chain rule. We establish that a sequence of locally Lipschitz maps between finite dimensional Euclidean spaces converges to a given locally Lipschitz map in the L-topology—that is, the weakest refinement of the sup norm topology on the space of locally Lipschitz maps that makes the generalised Jacobian a continuous functional—if and only if the limit superior of the sequence of directional derivatives of the maps in a given vector direction coincides with the generalised directional derivative of the given map in that direction, with the convergence to the limit superior being uniform for all unit vectors. We then prove our main result that the subspace of Lipschitz C ∞ maps between finite dimensional Euclidean spaces is dense in the space of Lipschitz maps equipped with the L-topology, and, for a given Lipschitz map, we explicitly construct a sequence of Lipschitz C ∞ maps converging to it in the L-topology, allowing global smooth approximation of a Lipschitz map and its differential properties. As an application, we obtain a short proof of the extension of Green’s theorem to interval-valued vector fields. For infinite dimensions, we show that the subgradient of a Lipschitz map on a Banach space is upper continuous, and, for a given real-valued Lipschitz map on a separable Banach space, we construct a sequence of Gateaux differentiable functions that converges to the map in the sup norm topology such that the limit superior of the directional derivatives in any direction coincides with the generalised directional derivative of the Lipschitz map in that direction.

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 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.

Quotients of Essentially Euclidean Spaces

2019 ◽  
Vol 62 (1) ◽  
pp. 71-74
Author(s):  
Tadeusz Figiel ◽  
William Johnson
Keyword(s):  
Normed Space ◽  
Quotient Space ◽  
Euclidean Spaces ◽  
Quantitative Version ◽  
Finite Dimensional ◽  
Dimensional Normed Space

AbstractA precise quantitative version of the following qualitative statement is proved: If a finite-dimensional normed space contains approximately Euclidean subspaces of all proportional dimensions, then every proportional dimensional quotient space has the same property.

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.

scholarly journals Littlewood-Paley theory on Gaussian spaces

1988 ◽  
Vol 109 ◽  
pp. 47-61 ◽  
Author(s):  
Jürgen Potthoff
Keyword(s):  
Lebesgue Measure ◽  
Measure Space ◽  
Euclidean Spaces ◽  
General Measure ◽  
Finite Dimensional

In this article we prove a number of inequalities of Littlewood-Paley-Stein (LPS) type for functions on general Gaussian spaces (s. below).In finite dimensional Euclidean spaces (with Lebesgue measure) the power of such inequalities has been demonstrated in Stein’s book [12]. In his second book [13], Stein treats other spaces too: also the situation of a general measure space (X, μ). However the latter case is too general to allow for a rich class of inequalities (cf. Theorem 10 in [13]).

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].

scholarly journals Metric Spaces Admitting Low-distortion Embeddings into All n-dimensional Banach Spaces

2016 ◽  
Vol 68 (4) ◽  
pp. 876-907 ◽  
Author(s):  
Mikhail Ostrovskii ◽  
Beata Randrianantoanina
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Metric Spaces ◽  
Euclidean Spaces ◽  
Dimensional Banach Space ◽  
Low Distortion ◽  
Bounded Distortion ◽  
Finite Metric Spaces ◽  
Uniformly Bounded

AbstractFor a fixed K > 1 and n ∈ ℕ, n ≫ 1, we study metric spaces which admit embeddings with distortion ≤ K into each n-dimensional Banach space. Classical examples include spaces embeddable into log n-dimensional Euclidean spaces, and equilateral spaces.We prove that good embeddability properties are preserved under the operation of metric composition of metric spaces. In particular, we prove that n-point ultrametrics can be embedded with uniformly bounded distortions into arbitrary Banach spaces of dimension log n.The main result of the paper is a new example of a family of finite metric spaces which are not metric compositions of classical examples and which do embed with uniformly bounded distortion into any Banach space of dimension n. This partially answers a question of G. Schechtman.

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