New Characterizations of the Reflexivity in Terms of the Set of Norm Attaining Functionals

1998 ◽  
Vol 41 (3) ◽  
pp. 279-289 ◽  
Author(s):  
Mariá D. Acosta ◽  
Manuel Ruiz Galán
Keyword(s):  
Banach Space ◽  
Positive Result ◽  
Unit Ball ◽  
Separable Banach Space ◽  
Intersection Property ◽  
Sufficient Condition ◽  
Empty Interior ◽  
Norm Topology ◽  
Fréchet Differentiable ◽  
Norm Attaining

AbstractAs a consequence of results due to Bourgain and Stegall, on a separable Banach space whose unit ball is not dentable, the set of norm attaining functionals has empty interior (in the norm topology). First we show that any Banach space can be renormed to fail this property. Then, our main positive result can be stated as follows: if a separable Banach space X is very smooth or its bidual satisfies the w*-Mazur intersection property, then either X is reflexive or the set of norm attaining functionals has empty interior, hence the same result holds if X has the Mazur intersection property and so, if the norm of X is Fréchet differentiable. However, we prove that smoothness is not a sufficient condition for the same conclusion.

scholarly journals On Gâteaux Differentiability of Pointwise Lipschitz Mappings

2008 ◽  
Vol 51 (2) ◽  
pp. 205-216 ◽  
Author(s):  
Jakub Duda
Keyword(s):  
Banach Space ◽  
Separable Banach Space ◽  
Monotone Function ◽  
Gâteaux Differentiability ◽  
Lipschitz Mappings ◽  
Gateaux Differentiability ◽  
Fréchet Differentiable ◽  
Set Of Points

AbstractWe prove that for every function f : X → Y , where X is a separable Banach space and Y is a Banach space with RNP, there exists a set A ∈ such that f is Gâteaux differentiable at all x ∈ S(f )\A, where S(f) is the set of points where f is pointwise-Lipschitz. This improves a result of Bongiorno. As a corollary, we obtain that every K-monotone function on a separable Banach space is Hadamard differentiable outside of a set belonging to ; this improves a result due to Borwein and Wang. Another corollary is that if X is Asplund, f : X → ℝ cone monotone, g : X → ℝ continuous convex, then there exists a point in X, where f is Hadamard differentiable and g is Fréchet differentiable.

scholarly journals The approximation problem for compact operators

1969 ◽  
Vol 1 (3) ◽  
pp. 397-401 ◽  
Author(s):  
S.R. Caradus
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Finite Rank ◽  
Reflexive Banach Space ◽  
Approximation Problem ◽  
Compact Operators ◽  
Sufficient Condition

The following sufficient condition is obtained for the uniform approximability of compact operators on a reflexive Banach space by operators of finite rank: if S is the unit ball of X and ø: X* → C(S) is the imbedding ø(x*)x = x*(x) then we require ø(X*) to be complemented in C(S).

INTEGRAL REPRESENTATIONS OF CYLINDRICAL LOCAL MARTINGALES IN EVERY SEPARABLE BANACH SPACE

2007 ◽  
Vol 10 (03) ◽  
pp. 365-379 ◽  
Author(s):  
MARTIN ONDREJÁT
Keyword(s):  
Banach Space ◽  
Wiener Process ◽  
Separable Banach Space ◽  
Stochastic Integral ◽  
Integral Representations ◽  
Local Martingale ◽  
Sufficient Condition ◽  
General Sufficient Condition

A general sufficient condition for a continuous cylindrical local martingale on a separable Banach space to be a stochastic integral with respect to a Wiener process is proven.

Smoothability, Strong Smoothability and Dentability in Banach Spaces1

1981 ◽  
Vol 24 (1) ◽  
pp. 59-68 ◽  
Author(s):  
R. Anantharaman ◽  
T. Lewis ◽  
J. H. M. Whitfield
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Asplund Space ◽  
Differentiable Norm ◽  
Nikodym Property ◽  
Fréchet Differentiable ◽  
Conjugate Banach Space

AbstractIt is shown that dentability of the unit ball of a conjugate Banach space X* does not imply smoothability of the unit ball of X, answering a question raised by Kemp. A property called strong smoothability is introduced and is shown to be dual to dentability. The results are used to provide new proofs of the facts that X is an Asplund space whenever it has an equivalent Fréchet differentiable norm, or whenever X* has the Radon-Nikodym Property.

scholarly journals Some questions about rotundity and renormings in Banach spaces

2005 ◽  
Vol 79 (1) ◽  
pp. 131-140 ◽  
Author(s):  
A. Aizpuru ◽  
F. J. Garcia-Pacheco
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Geometric Concepts ◽  
Strongly Exposed Points ◽  
Fréchet Differentiable ◽  
Rotund Banach Space

AbstractIn this paper, we show some results involving classical geometric concepts. For example, we characterize rotundity and Efimov-Stechkin property by mean of faces of the unit ball. Also, we prove that every almost locally uniformly rotund Banach space is locally uniformly rotund if its norm is Fréchet differentiable. Finally, we also provide some theorems in which we characterize the (strongly) exposed points of the unit ball using renormings.

TRANSLATING FINITE SETS INTO CONVEX SETS

2001 ◽  
Vol 33 (6) ◽  
pp. 711-714 ◽  
Author(s):  
EVA MATOUšKOVA
Keyword(s):  
Banach Space ◽  
Convex Subset ◽  
Separable Banach Space ◽  
Convex Sets ◽  
Reflexive Banach Space ◽  
Small Diameter ◽  
Empty Interior ◽  
Finite Sets ◽  
Bounded Set ◽  
Finite Set

Let X be a reflexive Banach space, and let C ⊂ X be a closed, convex and bounded set with empty interior. Then, for every δ > 0, there is a nonempty finite set F ⊂ X with an arbitrarily small diameter, such that C contains at most δ · |F| points of any translation of F. As a corollary, a separable Banach space X is reflexive if and only if every closed convex subset of X with empty interior is Haar null.

scholarly journals Geometric Constants in Banach Spaces Related to the Inscribed Quadrilateral of Unit Balls

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1294
Author(s):  
Asif Ahmad ◽  
Qi Liu ◽  
Yongjin Li
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Normal Structure ◽  
Inner Product ◽  
Sufficient Condition ◽  
Basic Properties ◽  
Geometric Constant ◽  
Geometric Constants

We introduce a new geometric constant Jin(X) based on a generalization of the parallelogram law, which is symmetric and related to the length of the inscribed quadrilateral side of the unit ball. We first investigate some basic properties of this new coefficient. Next, it is shown that, for a Banach space, Jin(X) becomes 16 if and only if the norm is induced by an inner product. Moreover, its properties and some relations between other well-known geometric constants are studied. Finally, a sufficient condition which implies normal structure is presented.

scholarly journals The Blum-Hanson Property

2019 ◽  
Vol 6 (1) ◽  
pp. 92-105
Author(s):  
Sophie Grivaux
Keyword(s):  
Banach Space ◽  
Hilbert Spaces ◽  
Separable Banach Space ◽  
Metric Spaces ◽  
Compact Metric Spaces ◽  
Theoretic Motivation

AbstractGiven a (real or complex, separable) Banach space, and a contraction T on X, we say that T has the Blum-Hanson property if whenever x, y ∈ X are such that Tnx tends weakly to y in X as n tends to infinity, the means{1 \over N}\sum\limits_{k = 1}^N {{T^{{n_k}}}x} tend to y in norm for every strictly increasing sequence (nk) k≥1 of integers. The space X itself has the Blum-Hanson property if every contraction on X has the Blum-Hanson property. We explain the ergodic-theoretic motivation for the Blum-Hanson property, prove that Hilbert spaces have the Blum-Hanson property, and then present a recent criterion of a geometric flavor, due to Lefèvre-Matheron-Primot, which allows to retrieve essentially all the known examples of spaces with the Blum-Hanson property. Lastly, following Lefèvre-Matheron, we characterize the compact metric spaces K such that the space C(K) has the Blum-Hanson property.

scholarly journals A note on best approximation and invertibility of operators on uniformly convex Banach spaces

1991 ◽  
Vol 14 (3) ◽  
pp. 611-614 ◽  
Author(s):  
James R. Holub
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Best Approximation ◽  
Bounded Linear Operator ◽  
Convex Banach Space ◽  
Sufficient Condition ◽  
Uniformly Convex ◽  
Bounded Linear ◽  
Uniformly Convex Banach Spaces

It is shown that ifXis a uniformly convex Banach space andSa bounded linear operator onXfor which‖I−S‖=1, thenSis invertible if and only if‖I−12S‖<1. From this it follows that ifSis invertible onXthen either (i)dist(I,[S])<1, or (ii)0is the unique best approximation toIfrom[S], a natural (partial) converse to the well-known sufficient condition for invertibility thatdist(I,[S])<1.

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.

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