scholarly journals Separability of Topological Groups: A Survey with Open Problems

Axioms ◽  
2018 ◽  
Vol 8 (1) ◽  
pp. 3 ◽  
Author(s):  
Arkady Leiderman ◽  
Sidney Morris
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Quotient Space ◽  
Topological Groups ◽  
Open Problems ◽  
Infinite Products ◽  
Matrix Groups ◽  
Compact Spaces ◽  
Finite Dimensional ◽  
Separable Quotient Problem

Separability is one of the basic topological properties. Most classical topological groups and Banach spaces are separable; as examples we mention compact metric groups, matrix groups, connected (finite-dimensional) Lie groups; and the Banach spaces C ( K ) for metrizable compact spaces K; and ℓ p , for p ≥ 1 . This survey focuses on the wealth of results that have appeared in recent years about separable topological groups. In this paper, the property of separability of topological groups is examined in the context of taking subgroups, finite or infinite products, and quotient homomorphisms. The open problem of Banach and Mazur, known as the Separable Quotient Problem for Banach spaces, asks whether every Banach space has a quotient space which is a separable Banach space. This paper records substantial results on the analogous problem for topological groups. Twenty open problems are included in the survey.

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 A characterisation of Hilbert spaces via orthogonality and proximinality

2005 ◽  
Vol 71 (1) ◽  
pp. 107-111
Author(s):  
Fathi B. Saidi
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Real Banach Space ◽  
Nonzero Element ◽  
Dimensional Subspace ◽  
Two Dimensional ◽  
Orthogonal Direction ◽  
Open Problems

In this paper we adopt the notion of orthogonality in Banach spaces introduced by the author in [6]. There, the author showed that in any two-dimensional subspace F of E, every nonzero element admits at most one orthogonal direction. The problem of existence of such orthogonal direction was not addressed before. Our main purpose in this paper is the investigation of this problem in the case where E is a real Banach space. As a result we obtain a characterisation of Hilbert spaces stating that, if in every two-dimensional subspace F of E every nonzero element admits an orthogonal direction, then E is isometric to a Hilbert space. We conclude by presenting some open problems.

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 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 ON A CHARACTERISTIC PROPERTY OF FINITE-DIMENSIONAL BANACH SPACES

2011 ◽  
Vol 53 (3) ◽  
pp. 443-449 ◽  
Author(s):  
ANTONÍN SLAVÍK
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Characteristic Property ◽  
Linear Operators ◽  
Infinite Dimensional ◽  
Counter Example ◽  
Finite Dimensional ◽  
Infinite Dimensional Banach Space ◽  
Dvoretzky’S Theorem ◽  
The Given

AbstractThis paper is inspired by a counter example of J. Kurzweil published in [5], whose intention was to demonstrate that a certain property of linear operators on finite-dimensional spaces need not be preserved in infinite dimension. We obtain a stronger result, which says that no infinite-dimensional Banach space can have the given property. Along the way, we will also derive an interesting proposition related to Dvoretzky's theorem.

scholarly journals Computability-theoretic complexity of effective Banach spaces

2022 ◽  
Author(s):  
◽  
Long Qian
Keyword(s):  
Banach Space ◽  
Lower Bound ◽  
Banach Spaces ◽  
Lower Bounds ◽  
Approximation Property ◽  
Upper Bounds ◽  
Space Theory ◽  
Approximation Properties ◽  
Open Problems

<p><b>We investigate the geometry of effective Banach spaces, namely a sequenceof approximation properties that lies in between a Banach space having a basis and the approximation property.</b></p> <p>We establish some upper bounds on suchproperties, as well as proving some arithmetical lower bounds. Unfortunately,the upper bounds obtained in some cases are far away from the lower bound.</p> <p>However, we will show that much tighter bounds will require genuinely newconstructions, and resolve long-standing open problems in Banach space theory.</p> <p>We also investigate the effectivisations of certain classical theorems in Banachspaces.</p>

Diametrically Maximal and Constant Width Sets in Banach Spaces

2006 ◽  
Vol 58 (4) ◽  
pp. 820-842 ◽  
Author(s):  
J. P. Moreno ◽  
P. L. Papini ◽  
R. R. Phelps
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Hausdorff Space ◽  
Constant Width ◽  
Negative Answer ◽  
Empty Interior ◽  
Finite Dimensional ◽  
Jung Constant

AbstractWe characterize diametrically maximal and constant width sets inC(K), whereKis any compact Hausdorff space. These results are applied to prove that the sum of two diametrically maximal sets needs not be diametrically maximal, thus solving a question raised in a paper by Groemer. A characterization of diametrically maximal sets inis also given, providing a negative answer to Groemer's problem in finite dimensional spaces. We characterize constant width sets inc0(I), for everyI, and then we establish the connections between the Jung constant of a Banach space and the existence of constant width sets with empty interior. Porosity properties of families of sets of constant width and rotundity properties of diametrically maximal sets are also investigated. Finally, we present some results concerning non-reflexive and Hilbert spaces.

FINITE DIMENSIONAL APPROXIMATIONS OF DIFFUSION PROCESSES ON BANACH SPACES

2001 ◽  
Vol 04 (04) ◽  
pp. 521-531
Author(s):  
T. S. ZHANG
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Quadratic Forms ◽  
Diffusion Processes ◽  
Existence Proof ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Dimensional Diffusion ◽  
Weak Limits

In this paper, we prove that the laws of diffusion processes on a Banach space X associated with quadratic forms (not necessarily symmetric) are the weak limits of laws of finite dimensional diffusions. A new existence proof for the infinite dimensional diffusion processes is obtained as a by-product.

Auerbach's theorem and tensor products of Banach spaces

1962 ◽  
Vol 58 (3) ◽  
pp. 476-480 ◽  
Author(s):  
A. F. Ruston
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Real Banach Space ◽  
Tensor Products ◽  
Finite Dimensional ◽  
Orthogonal Systems

In his well-known treatise, Banach mentions that there exist complete normed bi-orthogonal systems in any finite-dimensional (real) Banach space—a result he attributes to H. Auerbach ((1), p. 238). The purpose of this note is to present a proof of this result (valid for complex as well as real Banach spaces), and to give some applications to the theory of tensor products of Banach spaces.

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.

Sign in / Sign up

Export Citation Format

Share Document