Polar topology

In functional analysis and related areas of mathematics a polar topology, topology of -convergence or topology of uniform convergence on the sets of is a method to define locally convex topologies on the vector spaces of a pairing. Dual system A pairing is a triple consisting of two vector spaces over a field (either the real numbers or complex numbers) and a bilinear map A dual pair or dual system is a pairing satisfying the following two separation axioms: separates/distinguishes points of : for all non-zero there exists such that and separates/distinguishes points of : for all non-zero there exists such that Polar set The polar or absolute polar of a subset is the set Dually, the polar or absolute polar of a subset is denoted by and defined by In this case, the absolute polar of a subset is also called the prepolar of and may be denoted by The polar is a convex balanced set containing the origin. If then the bipolar of denoted by is defined by Similarly, if then the bipolar of is defined to be Suppose that is a pairing of vector spaces over Notation: For all let denote the linear functional on defined by and let Similarly, for all let be defined by and let The weak topology on induced by (and ) is the weakest TVS topology on denoted by or simply making all maps continuous, as ranges over Similarly, there are the dual definition of the weak topology on induced by (and ), which is denoted by or simply : it is the weakest TVS topology on making all maps continuous, as ranges over It is because of the following theorem that it is almost always assumed that the family consists of -bounded subsets of Every pairing can be associated with a corresponding pairing where by definition There is a repeating theme in duality theory, which is that any definition for a pairing has a corresponding dual definition for the pairing Convention and Definition: Given any definition for a pairing one obtains a dual definition by applying it to the pairing If the definition depends on the order of and (e.g.

About this result

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 (37)

Related people (8)

Related units (8)

Related concepts (12)

Related courses (3)

Related lectures (19)

MATH-502: Distribution and interpolation spaces

The goal of this course is to give an introduction to the theory of distributions and cover the fundamental results of Sobolev spaces including fractional spaces that appear in the interpolation theor

MATH-688: Reading group in applied topology I

The focus of this reading group is to delve into the concept of the "Magnitude of Metric Spaces". This approach offers an alternative approach to persistent homology to describe a metric space across

EE-465: Industrial electronics I

The course deals with the control of grid connected power electronic converters for renewable applications, covering: converter topologies, pulse width modulation, modelling, control algorithms and co

Optimal containment control for a class of heterogeneous multi-agent systems with actuator faults

Ju Wu, Tong Wang

This article investigates the optimal containment control problem for a class of heterogeneous multi-agent systems with time-varying actuator faults and unmatched disturbances based on adaptive dynamic programming. Since there exist unknown input signals i ...

WILEY2023

Decentralized learning made easy with DecentralizePy

Anne-Marie Kermarrec, Rafael Pereira Pires, Akash Balasaheb Dhasade, Rishi Sharma, Milos Vujasinovic

Decentralized learning (DL) has gained prominence for its potential benefits in terms of scalability, privacy, and fault tolerance. It consists of many nodes that coordinate without a central server and exchange millions of parameters in the inherently ite ...

2023

Communication-efficient distributed training of machine learning models

Thijs Vogels

In this thesis, we explore techniques for addressing the communication bottleneck in data-parallel distributed training of deep learning models. We investigate algorithms that either reduce the size of the messages that are exchanged between workers, or th ...

EPFL2023

Complete topological vector space

In functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point towards which they all get closer. The notion of "points that get progressively closer" is made rigorous by or , which are generalizations of , while "point towards which they all get closer" means that this Cauchy net or filter converges to The notion of completeness for TVSs uses the theory of uniform spaces as a framework to generalize the notion of completeness for metric spaces.

Strong dual space

In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) is the continuous dual space of equipped with the strong (dual) topology or the topology of uniform convergence on bounded subsets of where this topology is denoted by or The coarsest polar topology is called weak topology. The strong dual space plays such an important role in modern functional analysis, that the continuous dual space is usually assumed to have the strong dual topology unless indicated otherwise.

Dual system

In mathematics, a dual system, dual pair, or duality over a field is a triple consisting of two vector spaces and over and a non-degenerate bilinear map . Duality theory, the study of dual systems, is part of functional analysis. It is separate and distinct to Dual-system Theory in psychology. Pairings A or pair over a field is a triple which may also be denoted by consisting of two vector spaces and over (which this article assumes is the field either of real numbers or the complex numbers ).

Normed Spaces MATH-502: Distribution and interpolation spaces

Covers normed spaces, dual spaces, Banach spaces, Hilbert spaces, weak and strong convergence, reflexive spaces, and the Hahn-Banach theorem.

CW Complexes MATH-323: Topology III - Homology

Covers the construction and properties of CW complexes, including weak topology and characteristic maps.

Topological Groups MATH-225: Topology

Covers the definition and examples of topological groups, focusing on actions of groups on spaces.