scholarly journals A convex-valued selection theorem with a non-separable Banach space

2018 ◽  
Vol 7 (2) ◽  
pp. 197-209
Author(s):  
Pascal Gourdel ◽  
Nadia Mâagli
Keyword(s):  
Banach Space ◽  
Metric Space ◽  
Separable Banach Space ◽  
Lower Semicontinuous ◽  
Constructive Proof ◽  
Selection Theorem ◽  
Finite Dimensional ◽  
Nonempty Convex

AbstractIn the spirit of Michael’s selection theorem [6, Theorem 3.1”’], we consider a nonempty convex-valued lower semicontinuous correspondence {\varphi:X\to 2^{Y}}. We prove that if φ has either closed or finite-dimensional images, then there admits a continuous single-valued selection, where X is a metric space and Y is a Banach space. We provide a geometric and constructive proof of our main result based on the concept of peeling introduced in this paper.

scholarly journals On continuity and selections of multifunctions

2002 ◽  
Vol 65 (3) ◽  
pp. 407-422
Author(s):  
Pandelis Dodos
Keyword(s):  
Banach Space ◽  
Polish Space ◽  
Lower Semicontinuous ◽  
Selection Theorem ◽  
Weakly Compact ◽  
Compact Subsets

The notions of a Baire-1 and a weak Baire-1 multifunction are defined and a striking analogy between Baire-1 multifunctions and classical Baire-1 functions is established. A selection theorem is presented which asserts that if X is a metrisable space, Y a Polish space and F: X → 2Y/{∅} a closed-valued, weak Baire-1 multifunction, then F admits a Baire-1 selection. Using the machinery developed we prove that if X is a Banach space with separable dual, then every weak* usco, defined on a completely metrisable space Z, which values are weakly* compact subsets of the dual, is norm lower semicontinuous on a dense Gδ set.

scholarly journals Existence theorems for differential inclusions with nonconvex right hand side

1986 ◽  
Vol 9 (3) ◽  
pp. 459-469 ◽  
Author(s):  
Nikolaos S. Papageorgiou
Keyword(s):  
Banach Space ◽  
Separable Banach Space ◽  
Differential Inclusions ◽  
Existence Theorems ◽  
Lower Semicontinuous ◽  
Time Variable ◽  
The State ◽  
Infinite Dimensional ◽  
State Variable ◽  
Right Hand

In this paper we prove some new existence theorems for differential inclusions with a nonconvex right hand side, which is lower semicontinuous or continuous in the state variable, measurable in the time variable and takes values in a finite or infinite dimensional separable Banach space.

scholarly journals DISTORTION IN THE FINITE DETERMINATION RESULT FOR EMBEDDINGS OF LOCALLY FINITE METRIC SPACES INTO BANACH SPACES

2018 ◽  
Vol 61 (1) ◽  
pp. 33-47 ◽  
Author(s):  
S. OSTROVSKA ◽  
M. I. OSTROVSKII
Keyword(s):  
Banach Space ◽  
Real Number ◽  
Banach Spaces ◽  
Metric Space ◽  
Metric Spaces ◽  
Universal Constant ◽  
Locally Finite ◽  
Finite Dimensional ◽  
Finite Metric Space ◽  
Finite Metric Spaces

AbstractGiven a Banach spaceXand a real number α ≥ 1, we write: (1)D(X) ≤ α if, for any locally finite metric spaceA, all finite subsets of which admit bilipschitz embeddings intoXwith distortions ≤C, the spaceAitself admits a bilipschitz embedding intoXwith distortion ≤ α ⋅C; (2)D(X) = α+if, for every ϵ > 0, the conditionD(X) ≤ α + ϵ holds, whileD(X) ≤ α does not; (3)D(X) ≤ α+ifD(X) = α+orD(X) ≤ α. It is known thatD(X) is bounded by a universal constant, but the available estimates for this constant are rather large. The following results have been proved in this work: (1)D((⊕n=1∞Xn)p) ≤ 1+for every nested family of finite-dimensional Banach spaces {Xn}n=1∞and every 1 ≤p≤ ∞. (2)D((⊕n=1∞ℓ∞n)p) = 1+for 1 <p< ∞. (3)D(X) ≤ 4+for every Banach spaceXwith no nontrivial cotype. Statement (3) is a strengthening of the Baudier–Lancien result (2008).

scholarly journals Banach Spaces of Almost Universal Complemented Disposition

2019 ◽  
Vol 71 (1) ◽  
pp. 139-174
Author(s):  
Jesús M F Castillo ◽  
Yolanda Moreno
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Separable Banach Space ◽  
Complemented Subspace ◽  
Finite Dimensional ◽  
Dimensional Decomposition ◽  
Finite Dimensional Decomposition ◽  
Separable Spaces

Abstract We introduce and study the notion of space of almost universal complemented disposition (a.u.c.d.) as a generalization of Kadec space. We show that every Banach space with separable dual is isometrically contained as a $1$-complemented subspace of a separable a.u.c.d. space and that all a.u.c.d. spaces with $1$-finite dimensional decomposition (FDD) are isometric and contain isometric $1$-complemented copies of every separable Banach space with $1$-FDD. We then study spaces of universal complemented disposition (u.c.d.) and provide different constructions for such spaces. We also consider spaces of u.c.d. with respect to separable spaces.

scholarly journals Phelps spaces and finite dimensional decompositions

1988 ◽  
Vol 37 (2) ◽  
pp. 263-271 ◽  
Author(s):  
R. Deville ◽  
G. Godefroy ◽  
D.E.G. Hare ◽  
V. Zizler
Keyword(s):  
Banach Space ◽  
Separable Banach Space ◽  
Continuity Property ◽  
Finite Dimensional ◽  
Point Of Continuity Property ◽  
Convex Point

We show that if X is a separable Banach space such that X* fails the weak* convex point-of-continuity property (C*PCP), then there is a subspace Y of X such that both Y* and (X/Y)* fail C*PCP and both Y and X/Y have finite dimensional Schauder decompositions.

Separation and Weak König's Lemma

1999 ◽  
Vol 64 (1) ◽  
pp. 268-278 ◽  
Author(s):  
A. James Humphreys ◽  
Stephen G. Simpson
Keyword(s):  
Banach Space ◽  
Separable Banach Space ◽  
Convex Sets ◽  
Separation Theorem ◽  
Space Theory ◽  
Open Convex ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Weak König’S Lemma ◽  
König’S Lemma

AbstractWe continue the work of [14, 3, 1, 19, 16, 4, 12, 11, 20] investigating the strength of set existence axioms needed for separable Banach space theory. We show that the separation theorem for open convex sets is equivalent to WKL0 over RCA0. We show that the separation theorem for separably closed convex sets is equivalent to ACA0 over RCA0. Our strategy for proving these geometrical Hahn–Banach theorems is to reduce to the finite-dimensional case by means of a compactness argument.

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.

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.

scholarly journals On the unique predual problem for Lipschitz spaces

2017 ◽  
Vol 165 (3) ◽  
pp. 467-473 ◽  
Author(s):  
NIK WEAVER
Keyword(s):  
Banach Space ◽  
Metric Space ◽  
Lipschitz Spaces

AbstractFor any metric space X, the predual of Lip(X) is unique. If X has finite diameter or is complete and convex—in particular, if it is a Banach space—then the predual of Lip0(X) is unique.

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