Metric Spaces and Hilbert Spaces

2014 ◽  
pp. 381-416
Keyword(s):  
Hilbert Spaces ◽  
Metric Spaces

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 On the Metric Compactification of Infinite-dimensional Spaces

2018 ◽  
Vol 62 (3) ◽  
pp. 491-507 ◽  
Author(s):  
Armando W. Gutiérrez
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Metric Spaces ◽  
Infinite Dimensional ◽  
Full Characterization ◽  
Geodesic Metric

AbstractThe notion of metric compactification was introduced by Gromov and later rediscovered by Rieffel. It has been mainly studied on proper geodesic metric spaces. We present here a generalization of the metric compactification that can be applied to infinite-dimensional Banach spaces. Thereafter we give a complete description of the metric compactification of infinite-dimensional $\ell _{p}$ spaces for all $1\leqslant p<\infty$. We also give a full characterization of the metric compactification of infinite-dimensional Hilbert spaces.

scholarly journals Equivalence and stability of random fixed point iterative procedures

2006 ◽  
Vol 2006 ◽  
pp. 1-19 ◽  
Author(s):  
Ismat Beg ◽  
Mujahid Abbas
Keyword(s):  
Fixed Point ◽  
Hilbert Spaces ◽  
Metric Spaces ◽  
Necessary Conditions ◽  
Random Operator ◽  
Random Operators ◽  
Random Fixed Point ◽  
The Stability ◽  
Iterative Procedures ◽  
Measurable Mappings

We generate a sequence of measurable mappings iteratively and study necessary conditions for its strong convergence to a random fixed point of strongly pseudocontractive random operator. We establish the weak convergence of an implicit random iterative procedure to common random fixed point of a finite family of nonexpansive random operators in Hilbert spaces. We prove the equivalence between the convergence of random Ishikawa and random Mann iterative schemes for contraction random operator and strongly pseudocontractive random operator. We also examine the stability of random fixed point iterative procedures for the random operators satisfying certain contractive conditions in the context of metric spaces.

Uniform embeddings of metric spaces and of banach spaces into hilbert spaces

1985 ◽  
Vol 52 (3) ◽  
pp. 251-265 ◽  
Author(s):  
I. Aharoni ◽  
B. Maurey ◽  
B. S. Mityagin
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Metric Spaces

scholarly journals Fixed point iterations for Prešić-Kannan nonexpansive mappings in product convex metric spaces

2018 ◽  
Vol 10 (1) ◽  
pp. 56-69
Author(s):  
Hafiz Fukhar-ud-din ◽  
Vasile Berinde
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Metric Spaces ◽  
Unique Fixed Point ◽  
Nonexpansive Mappings ◽  
Uniformly Convex ◽  
Product Spaces ◽  
Convex Metric Spaces ◽  
Fixed Point Iterations

Abstract We introduce Prešić-Kannan nonexpansive mappings on the product spaces and show that they have a unique fixed point in uniformly convex metric spaces. Moreover, we approximate this fixed point by Mann iterations. Our results are new in the literature and are valid in Hilbert spaces, CAT(0) spaces and Banach spaces simultaneously.

scholarly journals Determining Fuzzy Distance between Sets by Application of Fixed Point Technique Using Weak Contractions and Fuzzy Geometric Notions

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 812
Author(s):  
Parbati Saha ◽  
Shantau Guria ◽  
Binayak S. Choudhury ◽  
Manuel De la Sen
Keyword(s):  
Fixed Point ◽  
Hilbert Spaces ◽  
Metric Spaces ◽  
Point Problem ◽  
Optimal Approximate Solution ◽  
Weak Contraction ◽  
Fuzzy Distance ◽  
Fixed Point Equation ◽  
Point Equation ◽  
Fixed Point Technique

In the present paper, we solve the problem of determining the fuzzy distance between two subsets of a fuzzy metric space. We address the problem by reducing it to the problem of finding an optimal approximate solution of a fixed point equation. This approach is well studied for the corresponding problem in metric spaces and is known as proximity point problem. We employ fuzzy weak contractions for that purpose. Fuzzy weak contraction is a recently introduced concept intermediate to a fuzzy contraction and a fuzzy non-expansive mapping. Fuzzy versions of some geometric properties essentially belonging to Hilbert spaces are considered in the main theorem. We include an illustrative example and two corollaries, one of which comes from a well-known fixed point theorem. The illustrative example shows that the main theorem properly includes its corollaries. The work is in the domain of fuzzy global optimization by use of fixed point methods.

On Some Applications of Graph Theory III

1972 ◽  
Vol 15 (1) ◽  
pp. 27-32 ◽  
Author(s):  
P. Erdös ◽  
A. Meir ◽  
V. T. Sós ◽  
P. Turán
Keyword(s):  
Graph Theory ◽  
Hilbert Spaces ◽  
Metric Spaces ◽  
Distance Distribution ◽  
Transfinite Diameter ◽  
Compact Sets ◽  
Basic Tool ◽  
Space Applications ◽  
Totally Bounded ◽  
Logical Result

In the first and second parts of this sequence we dealt with applications of graph theory to distance distribution in certain sets in euclidean spaces, to potential theory, to estimations of the transfinite diameter [1] and to value distribution of "triangle functional " (e.g. perimeter, area of triangles) [2]. The basic tool is provided in all these applications by the result formulated as Lemma 2. This, an essentially pure logical result, proves to be a very flexible and versatile instrument in applications.Here the same method is used in an abstract setting. First we deduce certain results for the density of a given family of subsets of an abstract set S in another family of subsets of the same S. Then we apply the results obtained to distance distribution in certain (e.g. totally bounded or compact) sets in metric spaces, in particular in a normed linear function space. Applications of this method to functional on Hilbert spaces were given by Katona [3].

Embedding quasi-metric spaces in Hilbert spaces

1982 ◽  
Vol 24 (1) ◽  
pp. 39-46
Author(s):  
Pham Son
Keyword(s):  
Hilbert Spaces ◽  
Metric Spaces
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