Concept# Metric space

Summary

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry.
The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another.
Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance and therefore

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications

Loading

Related people

Loading

Related units

Loading

Related concepts

Loading

Related courses

Loading

Related lectures

Loading

Related publications (47)

Loading

Loading

Loading

Related units (4)

Related people (5)

Related lectures (80)

Related concepts (205)

Continuous function

In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This

Compact space

In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no

Vector space

In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars. Scal

Using arguments developed by De Giorgi in the 1950's, it is possible to prove the regularity of the solutions to a vast class of variational problems in the Euclidean space. The main goal of the present thesis is to extend these results to the more abstract context of metric spaces with a measure. In particular, working in the axiomatic framework of Gol'dshtein – Troyanov, we establish both the interior and the boundary regularity of quasi-minimizers of the p-Dirichlet energy. Our proof works for quite general domains, assuming some natural hypotheses on the (axiomatic) D-structure. Furthermore, we prove analogous results for extremal functions lying in the class of Sobolev functions in the sense of Hajłasz – Koskela, i.e. functions characterized by the single condition that a Poincaré inequality be satisfied. Our strategy to prove these regularity results is first to show that, in a very general setting, the (Hölder) continuity of a function is a consequence of three specific technical hypotheses. This part of the argument is the essence of the De Giorgi method. Then, we verify that for a function u which is a quasi-minimizer in an axiomatic Sobolev space or an extremal Sobolev function in the sense of Hajłasz – Koskela, these technical hypotheses are indeed satisfied and u is thus (Hölder) continuous. In addition to that, we establish the Harnack's inequality for these extremal functions, and we show that the Dirichlet semi-norm of a piecewise-extremal function is equivalent to the sum of the Dirichlet semi-norms of its components.

Related courses (39)

Concepts de base de l'analyse fonctionnelle linéaire: opérateurs bornés, opérateurs compacts, théorie spectrale pour les opérateurs symétriques et compacts, le théorème de Hahn-Banach, les théorèmes de l'application ouverte et du graphe fermé.

The aim of this course is to provide a solid foundation of theory of distributions, Sobolev spaces and an introduction to the more general theory of interpolation spaces.

The course aims to introduce the basic concepts and results on metric embeddings, or more precisely on approximate embeddings. This area has been under rapid development since the 90's and it has strong impact on algorithms for discrete optimization problems.

This work is dedicated to the study of Borel equivalence relations acting on Borel fields of CAT(0) metric spaces over a standard probability space. In this new framework we get similar results to some theorems proved recently by S. Adams-W. Ballmann or N. Monod concerning groups of isometries of CAT(0) spaces. In Chapter 1, we build several Borel structures on a variety of fields before dealing in particular with Borel fields of CAT(0) spaces. Chapter 2 discusses the notion of an action for an equivalence relation on a field of metric spaces and gives several examples. We also introduce a definition of amenability for equivalence relations in terms of invariant section following an idea of R.J. Zimmer. Chapter 3 deals with the action of an amenable equivalence relation and shows that such a relation cannot act without fixing a section at infinity or preserving a subfield of Euclidean spaces. In Chapter 4, we show that if an equivalence relation is generated by two commuting groups and acts without fixing a section at infinity, then the field splits equivariantly and isometrically as a product. Using this result we also show that equivalence relations containing two coamenable subrelations cannot act without fixing a section at infinity or preserving a subfield of Euclidean spaces.