scholarly journals Tight Embeddability of Proper and Stable Metric Spaces

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
F. Baudier ◽  
G. Lancien
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Metric Space ◽  
Metric Spaces ◽  
Proper Subset ◽  
Reflexive Banach Spaces

Abstract We introduce the notions of almost Lipschitz embeddability and nearly isometric embeddability. We prove that for p ∈ [1,∞], every proper subset of Lp is almost Lipschitzly embeddable into a Banach space X if and only if X contains uniformly the ℓpn’s. We also sharpen a result of N. Kalton by showing that every stable metric space is nearly isometrically embeddable in the class of reflexive Banach spaces.

scholarly journals Expectation in metric spaces and characterizations of Banach spaces

2009 ◽  
Vol 42 (4) ◽  
Author(s):  
Artur Bator ◽  
Wiesław Zięba
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Metric Space ◽  
Metric Spaces ◽  
Bochner Integral ◽  
Random Elements

AbstractWe consider different definitions of expectation of random elements taking values in metric spaces. All such definitions are valid also in Banach spaces and in this case the results coincide with the Bochner integral. There may exist an isometry between considered metric space and some Banach space and in this case one can use the Bochner integral instead of expectation in metric space. We give some conditions which ensure existence of such isometry, for two different definitions of expectation in metric 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 The ideal of Lipschitz classical p-compact operators and its injective hull

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Toufik Tiaiba ◽  
Dahmane Achour
Keyword(s):  
Banach Space ◽  
Compact Operator ◽  
Metric Space ◽  
Metric Spaces ◽  
Compact Operators ◽  
Operator Ideal ◽  
Linear Case ◽  
Injective Hull ◽  
Nuclear Operators ◽  
The Ideal

Abstract We introduce and investigate the injective hull of the strongly Lipschitz classical p-compact operator ideal defined between a pointed metric space and a Banach space. As an application we extend some characterizations of the injective hull of the strongly Lipschitz classical p-compact from the linear case to the Lipschitz case. Also, we introduce the ideal of Lipschitz unconditionally quasi p-nuclear operators between pointed metric spaces and show that it coincides with the Lipschitz injective hull of the ideal of Lipschitz classical p-compact operators.

scholarly journals Almost Surjective Epsilon-Isometry in The Reflexive Banach Spaces

CAUCHY ◽  
2017 ◽  
Vol 4 (4) ◽  
pp. 167
Author(s):  
Minanur Rohman
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Lorentz Space ◽  
Bounded Linear Operator ◽  
Closed Subspace ◽  
Reflexive Banach Space ◽  
Reflexive Banach Spaces ◽  
Bounded Linear ◽  
Star Topology

<p class="AbstractCxSpFirst">In this paper, we will discuss some applications of almost surjective epsilon-isometry mapping, one of them is in Lorentz space ( L_(p,q)-space). Furthermore, using some classical theorems of w star-topology and concept of closed subspace -complemented, for every almost surjective epsilon-isometry mapping  <em>f </em>: <em>X to</em><em> Y</em>, where <em>Y</em> is a reflexive Banach space, then there exists a bounded linear operator   <em>T</em> : <em>Y to</em><em> X</em>  with  such that</p><p class="AbstractCxSpMiddle">  </p><p class="AbstractCxSpLast">for every x in X.</p>

scholarly journals Metric Spaces Admitting Low-distortion Embeddings into All n-dimensional Banach Spaces

2016 ◽  
Vol 68 (4) ◽  
pp. 876-907 ◽  
Author(s):  
Mikhail Ostrovskii ◽  
Beata Randrianantoanina
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Metric Spaces ◽  
Euclidean Spaces ◽  
Dimensional Banach Space ◽  
Low Distortion ◽  
Bounded Distortion ◽  
Finite Metric Spaces ◽  
Uniformly Bounded

AbstractFor a fixed K > 1 and n ∈ ℕ, n ≫ 1, we study metric spaces which admit embeddings with distortion ≤ K into each n-dimensional Banach space. Classical examples include spaces embeddable into log n-dimensional Euclidean spaces, and equilateral spaces.We prove that good embeddability properties are preserved under the operation of metric composition of metric spaces. In particular, we prove that n-point ultrametrics can be embedded with uniformly bounded distortions into arbitrary Banach spaces of dimension log n.The main result of the paper is a new example of a family of finite metric spaces which are not metric compositions of classical examples and which do embed with uniformly bounded distortion into any Banach space of dimension n. This partially answers a question of G. Schechtman.

scholarly journals SPACES OF SMALL METRIC COTYPE

2010 ◽  
Vol 02 (04) ◽  
pp. 581-597 ◽  
Author(s):  
E. VEOMETT ◽  
K. WILDRICK
Keyword(s):  
Banach Space ◽  
Metric Space ◽  
Metric Spaces ◽  
Ultrametric Space ◽  
Partial Converse ◽  
Definition Of ◽  
Hausdorff Limits

Mendel and Naor's definition of metric cotype extends the notion of the Rademacher cotype of a Banach space to all metric spaces. Every Banach space has metric cotype at least 2. We show that any metric space that is bi-Lipschitz is equivalent to an ultrametric space having infimal metric cotype 1. We discuss the invariance of metric cotype inequalities under snowflaking mappings and Gromov–Hausdorff limits, and use these facts to establish a partial converse of the main result.

scholarly journals Factoring absolutely summing operators through Hilbert-Schmidt operators

1989 ◽  
Vol 31 (2) ◽  
pp. 131-135 ◽  
Author(s):  
Hans Jarchow
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Hausdorff Space ◽  
Bounded Operator ◽  
Continuous Functions ◽  
Space Of Continuous Functions ◽  
Reflexive Banach Spaces ◽  
Summing Operator ◽  
Schmidt Operator ◽  
Second Dual

Let K be a compact Hausdorff space, and let C(K) be the corresponding Banach space of continuous functions on K. It is well-known that every 1-summing operator S:C(K)→l2 is also nuclear, and therefore factors S = S1S2, with S1:l2→l2 a Hilbert–Schmidt operator and S1:C(K)→l2 a bounded operator. It is easily seen that this latter property is preserved when C(K) is replaced by any quotient, and that a Banach space X enjoys this property if and only if its second dual, X**, does. This led A. Pełczyński [15] to ask if the second dual of a Banach space X must be isomorphic to a quotient of a C(K)-space if X has the property that every 1-summing operator X-→l2 factors through a Hilbert-Schmidt operator. In this paper, we shall first of all reformulate the question in an appropriate manner and then show that counter-examples are available among super-reflexive Tsirelson-like spaces as well as among quasi-reflexive Banach spaces.

scholarly journals Compact reduction in Lipschitz-free spaces

2021 ◽  
Vol 151 (6) ◽  
pp. 1683-1699
Author(s):  
Ramón J. Aliaga ◽  
Camille Noûs ◽  
Colin Petitjean ◽  
Antonín Procházka
Keyword(s):  
Banach Space ◽  
Metric Space ◽  
General Principle ◽  
Metric Spaces ◽  
Approximation Property ◽  
Complete Metric Space ◽  
Infinite Dimensional ◽  
Schur Property ◽  
Free Spaces ◽  
Compact Subsets

We prove a general principle satisfied by weakly precompact sets of Lipschitz-free spaces. By this principle, certain infinite dimensional phenomena in Lipschitz-free spaces over general metric spaces may be reduced to the same phenomena in free spaces over their compact subsets. As easy consequences we derive several new and some known results. The main new results are: $\mathcal {F}(X)$ is weakly sequentially complete for every superreflexive Banach space $X$, and $\mathcal {F}(M)$ has the Schur property and the approximation property for every scattered complete metric space $M$.

scholarly journals Coarse infinite-dimensionality of hyperspaces of finite subsets

2021 ◽  
Author(s):  
Thomas Weighill ◽  
Takamitsu Yamauchi ◽  
Nicolò Zava
Keyword(s):  
Banach Space ◽  
Metric Space ◽  
Metric Spaces ◽  
Hausdorff Metric ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Dimensional Properties ◽  
Bounded Geometry ◽  
Infinite Dimensional

AbstractWe consider infinite-dimensional properties in coarse geometry for hyperspaces consisting of finite subsets of metric spaces with the Hausdorff metric. We see that several infinite-dimensional properties are preserved by taking the hyperspace of subsets with at most n points. On the other hand, we prove that, if a metric space contains a sequence of long intervals coarsely, then its hyperspace of finite subsets is not coarsely embeddable into any uniformly convex Banach space. As a corollary, the hyperspace of finite subsets of the real line is not coarsely embeddable into any uniformly convex Banach space. It is also shown that every (not necessarily bounded geometry) metric space with straight finite decomposition complexity has metric sparsification property.

scholarly journals Tykhonov triples, well-posedness and convergence results

2021 ◽  
Vol 37 (1) ◽  
pp. 135-143
Author(s):  
YI-BIN XIAO ◽  
MIRCEA SOFONEA
Keyword(s):  
Nonlinear Equation ◽  
Banach Spaces ◽  
Metric Spaces ◽  
Mathematical Object ◽  
Multivalued Function ◽  
Well Posedness ◽  
Unified Theory ◽  
Reflexive Banach Spaces ◽  
Preliminary Results ◽  
Convergence Results

"In this paper we present a unified theory of convergence results in the study of abstract problems. To this end we introduce a new mathematical object, the so-called Tykhonov triple $\cT=(I,\Omega,\cC)$, constructed by using a set of parameters $I$, a multivalued function $\Omega$ and a set of sequences $\cC$. Given a problem $\cP$ and a Tykhonov triple $\cT$, we introduce the notion of well-posedness of problem $\cP$ with respect to $\cT$ and provide several preliminary results, in the framework of metric spaces. Then we show how these abstract results can be used to obtain various convergences in the study of a nonlinear equation in reflexive Banach spaces. "

Sign in / Sign up

Export Citation Format

Share Document