scholarly journals REPRESENTATIONS OF IDEALS IN POLISH GROUPS AND IN BANACH SPACES

2015 ◽  
Vol 80 (4) ◽  
pp. 1268-1289 ◽  
Author(s):  
PIOTR BORODULIN–NADZIEJA ◽  
BARNABÁS FARKAS ◽  
GRZEGORZ PLEBANEK
Keyword(s):  
Banach Space ◽  
Abelian Group ◽  
Banach Spaces ◽  
Open Problems ◽  
Polish Groups ◽  
Polish Group ◽  
Focus On The Family ◽  
The Family ◽  
Null Ideal ◽  
Defining Sequence

AbstractWe investigate ideals of the form {A⊆ω: Σn∈Axnis unconditionally convergent} where (xn)n∈ωis a sequence in a Polish group or in a Banach space. If an ideal onωcan be seen in this form for some sequence inX, then we say that it is representable inX.After numerous examples we show the following theorems: (1) An ideal is representable in a Polish Abelian group iff it is an analytic P-ideal. (2) An ideal is representable in a Banach space iff it is a nonpathological analytic P-ideal.We focus on the family of ideals representable inc0. We characterize this property via the defining sequence of measures. We prove that the trace of the null ideal, Farah’s ideal, and Tsirelson ideals are not representable inc0, and that a tallFσP-ideal is representable inc0iff it is a summable ideal. Also, we provide an example of a peculiar ideal which is representable inℓ1but not in ℝ.Finally, we summarize some open problems of this topic.

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 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>

scholarly journals The algebraic size of the family of injective operators

2017 ◽  
Vol 15 (1) ◽  
pp. 13-20 ◽  
Author(s):  
Luis Bernal-González
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Linear Algebra ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Infinite Dimensional Banach Space ◽  
The Family ◽  
One To One ◽  
Algebra Of Operators

Abstract In this paper, a criterion for the existence of large linear algebras consisting, except for zero, of one-to-one operators on an infinite dimensional Banach space is provided. As a consequence, it is shown that every separable infinite dimensional Banach space supports a commutative infinitely generated free linear algebra of operators all of whose nonzero members are one-to-one. In certain cases, the assertion holds for nonseparable Banach spaces.

scholarly journals On the problem of non-smoothness of non-reflexive second conjugate spaces

1975 ◽  
Vol 12 (3) ◽  
pp. 407-416 ◽  
Author(s):  
Ivan Singer
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Linear Subspace ◽  
Canonical Embedding ◽  
Reflexive Banach Spaces ◽  
Closed Linear Subspace ◽  
Conjugate Space ◽  
The Family

We prove that if E is a Banach space which has a subspace G such that the conjugate space G* contains a proper norm closed linear subspace V of characteristic 1, then E** is not smooth and there exist in πE(E) points of non-smoothness for E**, where πE: E → E** is the canonical embedding. We show that the spaces E having such a subspace G constitute a large proper subfamily of the family of all non-reflexive Banach spaces.

Atomic systems for operators

2019 ◽  
Vol 17 (01) ◽  
pp. 1850066 ◽  
Author(s):  
Khole Timothy Poumai ◽  
Shah Jahan
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Atomic System ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Bessel Sequence ◽  
Atomic Systems ◽  
The Family ◽  
Necessary And Sufficient

Gavruta [L. Gavruta, Frames for operators, Appl. Comput. Harmon. Anal. 32 (2012) 139–144] introduced the notion of [Formula: see text]-frame and atomic system for an operator [Formula: see text] in Hilbert spaces. We extend these notions to Banach spaces and obtain various new results. A necessary and sufficient condition for the existence of an atomic system for an operator [Formula: see text] in a Banach space is given. Also, a characterization for the family of local atoms of subspaces of Banach spaces has been given. Further, we give methods to construct an atomic system for an operator [Formula: see text] from a given Bessel sequence and an [Formula: see text]-Bessel sequence. Finally, a result related to stability of atomic system for an operator [Formula: see text] in a Banach space has been given.

GENERALIZATION OF THE FRAME OPERATOR AND THE CANONICAL DUAL FRAME TO BANACH SPACES

2008 ◽  
Vol 01 (04) ◽  
pp. 631-643 ◽  
Author(s):  
Diana T. Stoeva
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Dual Frame ◽  
Frame Operator ◽  
Hilbert Frames ◽  
The Family

Xd-frames for Banach spaces are generalization of Hilbert frames. In this paper we extend the concepts of frame operator and canonical dual to the case of Xd-frames. For a given Xd-frame {gi} for the Banach space X we define an Xd-frame map𝕊 : X → X* and determine conditions, which imply that 𝕊 is invertible and the family {𝕊-1gi} is an [Formula: see text]-frame for X* such that f = ∑gi(f)𝕊-1gi for every f ∈ X and g = ∑g(𝕊-1gi)gi for every g ∈ X*. If X is a Hilbert space and {gi} is a frame for X, then the ℓ2-frame map 𝕊 gives the frame operator S and the family {𝕊-1gi} coincides with the canonical dual of {gi}.

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.

On the Hyers–Ulam stability and hyperstability of a generalized quadratic functional equation in n-Banach spaces

2021 ◽  
pp. 2250013
Author(s):  
Nordine Bounader
Keyword(s):  
Banach Space ◽  
Functional Equation ◽  
Abelian Group ◽  
Banach Spaces ◽  
Quadratic Functional Equation ◽  
Quadratic Functional ◽  
Ulam Stability ◽  
Class Of Functions

In this paper, we establish some Hyers–Ulam stability and hyperstability results of the following functional equation [Formula: see text] in the class of functions from an abelian group [Formula: see text] into a [Formula: see text]-Banach space.

ON HYPERSTABILITY OF ADDITIVE MAPPINGS ONTO BANACH SPACES

2014 ◽  
Vol 91 (2) ◽  
pp. 278-285 ◽  
Author(s):  
YUNBAI DONG ◽  
BENTUO ZHENG
Keyword(s):  
Banach Space ◽  
Abelian Group ◽  
Banach Spaces ◽  
Studia Math ◽  
Additive Map ◽  
Image Position ◽  
Additive Mappings

AbstractLet$(X,+)$be an Abelian group and$E$be a Banach space. Suppose that$f:X\rightarrow E$is a surjective map satisfying the inequality$$\begin{eqnarray}|\,\Vert f(x)-f(y)\Vert -\Vert f(x-y)\Vert \,|\leq {\it\varepsilon}\min \{\Vert f(x)-f(y)\Vert ^{p},\Vert f(x-y)\Vert ^{p}\}\end{eqnarray}$$for some${\it\varepsilon}>0$,$p>1$and for all$x,y\in X$. We prove that$f$is an additive map. However, this result does not hold for$0<p\leq 1$. As an application, we show that if$f$is a surjective map from a Banach space$E$onto a Banach space$F$so that for some${\it\epsilon}>0$and$p>1$$$\begin{eqnarray}|\,\Vert f(x)-f(y)\Vert -\Vert f(u)-f(v)\Vert \,|\leq {\it\epsilon}\min \{\Vert f(x)-f(y)\Vert ^{p},\Vert f(u)-f(v)\Vert ^{p}\}\end{eqnarray}$$whenever$\Vert x-y\Vert =\Vert u-v\Vert$, then$f$preserves equality of distance. Moreover, if$\dim E\geq 2$, there exists a constant$K\neq 0$such that$Kf$is an affine isometry. This improves a result of Vogt [‘Maps which preserve equality of distance’,Studia Math.45(1973) 43–48].

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>

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