Strict U-Ideals and U-Summands in Banach Spaces

2014 ◽  
Vol 114 (2) ◽  
pp. 216 ◽  
Author(s):  
Trond A. Abrahamsen
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Extension Operator ◽  
Denting Points ◽  
Set Of Points

For a strict u-ideal $X$ in a Banach space $Y$ we show that the set of points in the dual unit ball $B_{X^{\ast}}$, strongly exposed by points in the range $\it TY$ of the unconditional extension operator $T$ from $Y$ into the bidual $X^{\ast\ast}$ of $X$, is contained in the weak$^{\ast}$ denting points in $B_{X^{\ast}}$. We also prove that a u-embedded space is a u-summand if and only if it contains no copy of $c_0$ if and only if it is weakly sequentially complete.

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.

Banach Spaces that are Uniformly Rotund in Weakly Compact Sets of Directions

1977 ◽  
Vol 29 (5) ◽  
pp. 963-970 ◽  
Author(s):  
Mark A. Smith
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Nonzero Element ◽  
Bounded Subset ◽  
Compact Sets ◽  
Weakly Compact ◽  
Uniform Rotundity ◽  
Minimum Depth ◽  
Weakly Compact Sets

In a Banach space, the directional modulus of rotundity, δ (ϵ, z), measures the minimum depth at which the midpoints of all chords of the unit ball which are parallel to z and of length at least ϵ are buried beneath the surface. A Banach space is uniformly rotund in every direction (URED) if δ (ϵ, z) is positive for every positive ϵ and every nonzero element z. This concept of directionalized uniform rotundity was introduced by Garkavi [6] to characterize those Banach spaces in which every bounded subset has at most one Čebyšev center.

scholarly journals Semi-embeddings of Banach spaces which are hereditarily c0

1983 ◽  
Vol 26 (2) ◽  
pp. 163-167 ◽  
Author(s):  
L. Drewnowski
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Vector Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Continuous Linear Operator ◽  
Closed Unit Ball ◽  
Continuous Linear ◽  
Scattered Space ◽  
Complete Set

Following Lotz, Peck and Porta [9], a continuous linear operator from one Banach space into another is called a semi-embedding if it is one-to-one and maps the closed unit ball of the domain onto a closed (hence complete) set. (Below we shall allow the codomain to be an F-space, i.e., a complete metrisable topological vector space.) One of the main results established in [9] is that if X is a compact scattered space, then every semi-embedding of C(X) into another Banach space is an isomorphism ([9], Main Theorem, (a)⇒(b)).

scholarly journals NEAREST POINTS AND DELTA CONVEX FUNCTIONS IN BANACH SPACES

2015 ◽  
Vol 93 (2) ◽  
pp. 283-294
Author(s):  
JONATHAN M. BORWEIN ◽  
OHAD GILADI
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Functions ◽  
Closed Set ◽  
Nearest Point ◽  
Set Of Points ◽  
Nearest Points

Given a closed set$C$in a Banach space$(X,\Vert \cdot \Vert )$, a point$x\in X$is said to have a nearest point in$C$if there exists$z\in C$such that$d_{C}(x)=\Vert x-z\Vert$, where$d_{C}$is the distance of$x$from$C$. We survey the problem of studying the size of the set of points in$X$which have nearest points in$C$. We then turn to the topic of delta convex functions and indicate how it is related to finding nearest points.

scholarly journals Locally Lipschitz functions are generically pseudo-regular on separable Banach spaces

1993 ◽  
Vol 47 (2) ◽  
pp. 205-212 ◽  
Author(s):  
J.R. Giles ◽  
Scott Sciffer
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Lipschitz Function ◽  
Separable Banach Space ◽  
Locally Lipschitz Function ◽  
Locally Lipschitz Functions ◽  
Dini Derivative ◽  
Locally Lipschitz ◽  
Set Of Points ◽  
Clarke Derivative

For a locally Lipschitz function on a separable Banach space the set of points of Gâteaux differentiability is dense but not necessarily residual. However, the set of points where the upper Dini derivative and the Clarke derivative agree is residual. It follows immediately that the set of points of intermediate differentiability is also residual and the set of points where the function is Gâteaux but not strictly differentiable is of the first category.

NONEXPANSIVE BIJECTIONS TO THE UNIT BALL OF THE -SUM OF STRICTLY CONVEX BANACH SPACES

2018 ◽  
Vol 97 (2) ◽  
pp. 285-292 ◽  
Author(s):  
V. KADETS ◽  
O. ZAVARZINA
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Convex Banach Space ◽  
Strictly Convex ◽  
Strictly Convex Banach Space ◽  
Strictly Convex Banach Spaces

Extending recent results by Cascales et al. [‘Plasticity of the unit ball of a strictly convex Banach space’, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat.110(2) (2016), 723–727], we demonstrate that for every Banach space $X$ and every collection $Z_{i},i\in I$, of strictly convex Banach spaces, every nonexpansive bijection from the unit ball of $X$ to the unit ball of the sum of $Z_{i}$ by $\ell _{1}$ is an isometry.

scholarly journals Delta- and Daugavet points in Banach spaces

2020 ◽  
Vol 63 (2) ◽  
pp. 475-496
Author(s):  
T. A. Abrahamsen ◽  
R. Haller ◽  
V. Lima ◽  
K. Pirk
Keyword(s):  
Banach Space ◽  
Convex Hull ◽  
Unit Ball ◽  
Banach Spaces ◽  
Unit Sphere ◽  
Closed Convex Hull ◽  
Müntz Spaces ◽  
Daugavet Property ◽  
Diameter Two Property

AbstractA Δ-point x of a Banach space is a norm-one element that is arbitrarily close to convex combinations of elements in the unit ball that are almost at distance 2 from x. If, in addition, every point in the unit ball is arbitrarily close to such convex combinations, x is a Daugavet point. A Banach space X has the Daugavet property if and only if every norm-one element is a Daugavet point. We show that Δ- and Daugavet points are the same in L1-spaces, in L1-preduals, as well as in a big class of Müntz spaces. We also provide an example of a Banach space where all points on the unit sphere are Δ-points, but none of them are Daugavet points. We also study the property that the unit ball is the closed convex hull of its Δ-points. This gives rise to a new diameter-two property that we call the convex diametral diameter-two property. We show that all C(K) spaces, K infinite compact Hausdorff, as well as all Müntz spaces have this property. Moreover, we show that this property is stable under absolute sums.

Smoothness, Convexity, Porosity, and Separable Determination

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Functions ◽  
Equivalent Norm ◽  
Fréchet Differentiability ◽  
Smooth Norm ◽  
Frechet Differentiability ◽  
Porous Sets ◽  
Set Of Points

This chapter shows how spaces with separable dual admit a Fréchet smooth norm. It first considers a criterion of the differentiability of continuous convex functions on Banach spaces before discussing Fréchet smooth and nonsmooth renormings and Fréchet differentiability of convex functions. It then describes the connection between porous sets and Fréchet differentiability, along with the set of points of Fréchet differentiability of maps between Banach spaces. It also examines the concept of separable determination, the relevance of the σ‎-porous sets for differentiability and proves the existence of a Fréchet smooth equivalent norm on a Banach space with separable dual. The chapter concludes by explaining how one can show that many differentiability type results hold in nonseparable spaces provided they hold in separable ones.

EXTREME POINTS FOR COMBINATORIAL BANACH SPACES

2018 ◽  
Vol 61 (2) ◽  
pp. 487-500
Author(s):  
KEVIN BEANLAND ◽  
NOAH DUNCAN ◽  
MICHAEL HOLT ◽  
JAMES QUIGLEY
Keyword(s):  
Banach Space ◽  
Lower Bound ◽  
Unit Ball ◽  
Banach Spaces ◽  
Extreme Points ◽  
Explicit Construction ◽  
Stability Properties ◽  
Original Space ◽  
Regular Family

AbstractA norm ‖ċ‖ on c00 is called combinatorial if there is a regular family of finite subsets $\mathcal{F}$, so that $\|x\|=\sup_{F \in \mathcal{F}} \sum_{i \in F} |x(i)|$. We prove the set of extreme points of the ball of a combinatorial Banach space is countable. This extends a theorem of Shura and Trautman. The second contribution of this article is to exhibit many new examples of extreme points for the unit ball of dual Tsirelson's original space and give an explicit construction of an uncountable collection of extreme points of the ball of Tsirelson's original space. We also prove some stability properties of the intermediate norms used to define Tsirelson's space and give a lower bound of the stabilization function for these intermediate norms.

scholarly journals Geometric Characterization of Injective Banach Lattices

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 250
Author(s):  
Anatoly Kusraev ◽  
Semën Kutateladze
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Set Theory ◽  
Banach Lattice ◽  
Dual Space ◽  
Banach Lattices ◽  
Transfer Principle ◽  
Ordered Banach Spaces

This is a continuation of the authors’ previous study of the geometric characterizations of the preduals of injective Banach lattices. We seek the properties of the unit ball of a Banach space which make the space isometric or isomorphic to an injective Banach lattice. The study bases on the Boolean valued transfer principle for injective Banach lattices. The latter states that each such lattice serves as an interpretation of an AL-space in an appropriate Boolean valued model of set theory. External identification of the internal Boolean valued properties of the corresponding AL-spaces yields a characterization of injective Banach lattices among Banach spaces and ordered Banach spaces. We also describe the structure of the dual space and present some dual characterization of injective Banach lattices.

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