A general learning algorithm for functions between metric spaces

Frames for Metric Spaces

We make a systematic study of frames for metric spaces. We prove that every separable metric space admits a metric $\mathcal{M}_d$-frame. Through Lipschitz-free Banach spaces we show that there is a correspondence between frames for metric spaces and frames for subsets of Banach spaces. We derive some characterizations of metric frames. We also derive stability results for metric frames.

Generalized Hausdorff metric on $S_{b}$-metric space and some fixed point results

In this paper, a metric on $S_b$-metric space analogous to the Hausdorff metric has been introduced and some basic properties are obtained on multi-valued $S_b$-metric space. Further, the fundamental multi-valued contraction of Nadler(1962) has been extended to the $S_b$-metric space setting, and two results have been established. The entire study is supported by suitable examples.

Metrizability of Cone Metric Spaces Via Renorming the Banach Spaces

In this paper we show that by renorming an ordered Banach space, every cone P can be converted to a normal cone with constant K = 1 and consequently due to this approach every cone metric space is really a metric one and every theorem in metric space valid for cone metric space automatically.

TVS-cone metric spaces as a special case of metric spaces

There have been a number of generalizations of fixed point results to the so called TVS-cone metric spaces, based on a distance function that takes values in some cone with nonempty interior (solid cone) in some topological vector space. In this paper we prove that the TVS-cone metric space can be equipped with a family of mutually equivalent (usual) metrics such that the convergence (resp. property of being Cauchy sequence, contractivity condition) in TVS sense is equivalent to convergence (resp. property of being Cauchy sequence, contractivity condition) in all of these metrics. As a consequ…

Free spaces over some proper metric spaces

We prove that the Lipschitz-free space over a countable proper metric space is isometric to a dual space and has the metric approximation property. We also show that the Lipschitz-free space over a proper ultrametric space is isometric to the dual of a space which is isomorphic to c_0.

Metrizability of Cone Metric spaces

In 2007 H. Long-Guang and Z. Xian, [H. Long-Guang and Z. Xian, Cone Metric Spaces and Fixed Point Theorems of Contractive Mapping, J. Math. Anal. Appl., 322(2007), 1468-1476], generalized the concept of a metric space, by introducing cone metric spaces, and obtained some fixed point theorems for mappings satisfying certain contractive conditions. The main question was "Are cone metric spaces a real generalization of metric spaces?" Throughout this paper we answer the question in the negative, proving that every cone metric space is metrizable and the equivalent metric satisfies the same contra…

An Operator-Valued Kantorovich Metric on Complete Metric Spaces

The Kantorovich metric provides a way of measuring the distance between two Borel probability measures on a metric space. This metric has a broad range of applications from bioinformatics to image processing, and is commonly linked to the optimal transport problem in computer science. Noteworthy to this paper will be the role of the Kantorovich metric in the study of iterated function systems, which are families of contractive mappings on a complete metric space. When the underlying metric space is compact, it is well known that the space of Borel probability measures on this metric space, equ…

Complete metric spaces with property (Z) are length metric spaces

We prove that every complete metric space with property (Z) is a length space. These answers questions posed by García-Lirola, Procházka and Rueda Zoca, and by Becerra Guerrero, López-Pérez and Rueda Zoca, related to the structure of Lipschitz-free Banach spaces of metric spaces.

Concentration in unbounded metric spaces and algorithmic stability

We prove an extension of McDiarmid's inequality for metric spaces with unbounded diameter. To this end, we introduce the notion of the {\em subgaussian diameter}, which is a distribution-dependent refinement of the metric diameter. Our technique provides an alternative approach to that of Kutin and Niyogi's method of weakly difference-bounded functions, and yields nontrivial, dimension-free results in some interesting cases where the former does not. As an application, we give apparently the first generalization bound in the algorithmic stability setting that holds for unbounded loss functions…

Round and sleek subspaces of linear metric spaces and metric spaces

In the recent work [Metrically round and sleek metric spaces, \emph{The Journal of Analysis} (2022), pp 1--17], the authors proved some results on metrically round and sleek linear metric spaces and metric spaces. In continuation, the present article discusses more results on such spaces along with identification of round and sleek subsets of linear metric spaces and metric spaces in the subspace topology.

Embeddings of locally finite metric spaces into Banach spaces

We show that if X is a Banach space without cotype, then every locally finite metric space embeds metrically into X.

Quantum metric spaces of quantum maps

We show that any quantum family of maps from a non commutative space to a compact quantum metric space has a canonical quantum semi metric structure.

关于度量空间的等距普遍性

On isometric universality of spaces of metrics

本文研究了度量空间的等距普遍性，证明了具有丰富子空间的可度量空间上的有界度量空间和度量空间分别对特定类别的度量空间具有普遍性，并给出了紧致可数空间下的反例。