scholarly journals LIPSCHITZ RETRACTION OF FINITE SUBSETS OF HILBERT SPACES

2015 ◽  
Vol 93 (1) ◽  
pp. 146-151 ◽  
Author(s):  
LEONID V. KOVALEV
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Finite Subset ◽  
Metric Space ◽  
Isometric Embeddings ◽  
Nested Sequence

Finite subset spaces of a metric space $X$ form a nested sequence under natural isometric embeddings $X=X(1)\subset X(2)\subset \cdots \,$. We prove that this sequence admits Lipschitz retractions $X(n)\rightarrow X(n-1)$ when $X$ is a Hilbert space.

scholarly journals Lipschitz Retractions in Hadamard Spaces via Gradient Flow Semigroups

2016 ◽  
Vol 59 (4) ◽  
pp. 673-681 ◽  
Author(s):  
Miroslav Bačák ◽  
Leonid V. Kovalev
Keyword(s):  
Hilbert Space ◽  
Finite Subset ◽  
Metric Space ◽  
Hausdorff Metric ◽  
Gradient Flow ◽  
The Other ◽  
Dimensional Sphere ◽  
Hadamard Space ◽  
One Dimensional ◽  
The One

AbstractLet X(n), for n ∊ ℕ, be the set of all subsets of a metric space (X, d) of cardinality at most n. The set X(n) equipped with the Hausdorff metric is called a finite subset space. In this paper we are concerned with the existence of Lipschitz retractions r: X(n)→ X(n − 1) for n ≥ 2. It is known that such retractions do not exist if X is the one-dimensional sphere. On the other hand, Kovalev has recently established their existence if X is a Hilbert space, and he also posed a question as to whether or not such Lipschitz retractions exist when X is a Hadamard space. In this paper we answer the question in the positive.

scholarly journals COARSE EMBEDDINGS INTO A HILBERT SPACE, HAAGERUP PROPERTY AND POINCARÉ INEQUALITIES

2009 ◽  
Vol 01 (01) ◽  
pp. 87-100 ◽  
Author(s):  
ROMAIN TESSERA
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Metric Space ◽  
Unitary Representations ◽  
Poincaré Inequalities ◽  
Haagerup Property ◽  
Relative Property ◽  
Poincare Inequalities ◽  
Isometric Actions ◽  
Property T

We prove that a metric space does not coarsely embed into a Hilbert space if and only if it satisfies a sequence of Poincaré inequalities, which can be formulated in terms of (generalized) expanders. We also give quantitative statements, relative to the compression. In the equivariant context, our result says that a group does not have the Haagerup Property if and only if it has relative property T with respect to a family of probabilities whose supports go to infinity. We give versions of this result both in terms of unitary representations, and in terms of affine isometric actions on Hilbert spaces.

scholarly journals Rigid subsets in Euclidean and Hilbert spaces

1975 ◽  
Vol 20 (1) ◽  
pp. 66-72 ◽  
Author(s):  
Ludvik Janos
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Metric Space ◽  
Separable Hilbert Space ◽  
Euclidean Spaces ◽  
Point Set

AbstractA subset Y of a metric space (X, p) is called rigid if all the distances p(y1, y2) between points y1, y2 ∈ Y in Y are mutually different. The main purpose of this paper is to prove the existence of dense rigid subsets of cardinality c in Euclidean spaces En and in the separable Hilbert space l2. Some applications to abstract point set geometries are given and the connection with the theory of dimension is discussed.

Homeomorphic Images of Orthogonal Bases

2016 ◽  
Vol 31 ◽  
pp. 485-491
Author(s):  
M. Kebryaee ◽  
M. Radjabalipour
Keyword(s):  
Hilbert Space ◽  
Linear Algebra ◽  
Hilbert Spaces ◽  
Finite Subset ◽  
Sufficient Conditions ◽  
Orthogonal Basis ◽  
Riesz Bases ◽  
Orthonormal Bases ◽  
Homeomorphic Image ◽  
Necessary And Sufficient

Necessary and sufficient conditions are obtained for a sequence $\{x_j:~j\in \mathbb J\}$ in a Hilbert space to be, up to the elimination of a finite subset of $\mathbb J$, the linear homeomorphic image of an orthogonal basis of some Hilbert space $K$. This extends a similar result for orthonormal bases due to Holub [J.R. Holub. Pre-frame operators, Besselian frames, and near-Riesz bases in Hilbert spaces. \textit{Proc. Amer. Math. Soc.}, 122(3):779--785, 1994]. The proofs given here are based on simple linear algebra techniques.

scholarly journals Translation and modulation invariant Hilbert spaces

2021 ◽  
Author(s):  
Joachim Toft ◽  
Anupam Gumber ◽  
Ramesh Manna ◽  
P. K. Ratnakumar
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Zero Element ◽  
Multiplicative Constant ◽  
Space Of Distributions ◽  
Feichtinger Algebra

AbstractLet $$\mathcal H$$ H be a Hilbert space of distributions on $$\mathbf{R}^{d}$$ R d which contains at least one non-zero element of the Feichtinger algebra $$S_0$$ S 0 and is continuously embedded in $$\mathscr {D}'$$ D ′ . If $$\mathcal H$$ H is translation and modulation invariant, also in the sense of its norm, then we prove that $$\mathcal H= L^2$$ H = L 2 , with the same norm apart from a multiplicative constant.

scholarly journals Clock-dependent spacetime

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Sung-Sik Lee
Keyword(s):  
Hilbert Space ◽  
General Relativity ◽  
Quantum Gravity ◽  
Hilbert Spaces ◽  
Coordinate Systems ◽  
Physical Laws

Abstract Einstein’s theory of general relativity is based on the premise that the physical laws take the same form in all coordinate systems. However, it still presumes a preferred decomposition of the total kinematic Hilbert space into local kinematic Hilbert spaces. In this paper, we consider a theory of quantum gravity that does not come with a preferred partitioning of the kinematic Hilbert space. It is pointed out that, in such a theory, dimension, signature, topology and geometry of spacetime depend on how a collection of local clocks is chosen within the kinematic Hilbert space.

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 Some Hilbert spaces related with the Dirichlet space

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Nicola Arcozzi ◽  
Pavel Mozolyako ◽  
Karl-Mikael Perfekt ◽  
Stefan Richter ◽  
Giulia Sarfatti
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Dirichlet Space

AbstractWe study the reproducing kernel Hilbert space with kernel k

scholarly journals Isometric Group Actions on Hilbert Spaces: Structure of Orbits

2008 ◽  
Vol 60 (5) ◽  
pp. 1001-1009 ◽  
Author(s):  
Yves de Cornulier ◽  
Romain Tessera ◽  
Alain Valette
Keyword(s):  
Hilbert Space ◽  
Nilpotent Group ◽  
Hilbert Spaces ◽  
Metabelian Group ◽  
Isometric Action ◽  
Finitely Generated ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Isometric Group Actions ◽  
Dense Orbits

AbstractOur main result is that a finitely generated nilpotent group has no isometric action on an infinite-dimensional Hilbert space with dense orbits. In contrast, we construct such an action with a finitely generated metabelian group.

scholarly journals On invex programming problem in Hilbert spaces

2015 ◽  
Vol 25 (3) ◽  
pp. 379-385
Author(s):  
Sandip Chatterjee ◽  
Rathindranath Mukherjee
Keyword(s):  
Hilbert Space ◽  
Programming Problem ◽  
Hilbert Spaces

In this paper we introduce the invex programming problem in Hilbert space. The requisite theory has been established to characterize the solution of such class of problems.

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