Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Weak topology
Summary
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a topological vector space (such as a normed vector space) with respect to its continuous dual. The remainder of this article will deal with this case, which is one of the concepts of functional analysis.
One may call subsets of a topological vector space weakly closed (respectively, weakly compact, etc.) if they are closed (respectively, compact, etc.) with respect to the weak topology. Likewise, functions are sometimes called weakly continuous (respectively, weakly differentiable, weakly analytic, etc.) if they are continuous (respectively, differentiable, analytic, etc.) with respect to the weak topology.
Starting in the early 1900s, David Hilbert and Marcel Riesz made extensive use of weak convergence. The early pioneers of functional analysis did not elevate norm convergence above weak convergence and oftentimes viewed weak convergence as preferable. In 1929, Banach introduced weak convergence for normed spaces and also introduced the analogous weak-* convergence. The weak topology is also called topologie faible and schwache Topologie.
Topologies on spaces of linear maps
Let be a topological field, namely a field with a topology such that addition, multiplication, and division are continuous. In most applications will be either the field of complex numbers or the field of real numbers with the familiar topologies.
Dual system#Weak topology
Both the weak topology and the weak* topology are special cases of a more general construction for pairings, which we now describe.
The benefit of this more general construction is that any definition or result proved for it applies to both the weak topology and the weak* topology, thereby making redundant the need for many definitions, theorem statements, and proofs.
Official source
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.
Ontological neighbourhood
Related courses (31)
MATH-726: Working group in Topology I
The theme of the working group varies from year to year. Examples of recent topics studied include: Galois theory of ring spectra, duality in algebra and topology, and topological algebraic geometry.
MATH-220: Metric and topological spaces
A topological space is a space endowed with a notion of nearness. A metric space is an example of a topological space, where the concept of nearness is measured by a distance function. Within this abs
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
Related lectures (90)
CW Complexes
Covers the construction and properties of CW complexes, including weak topology and characteristic maps.
Definition of Sobolew Spaces
Explains the definition of Sobolew spaces and their main properties, focusing on weak denivelre.
Seifert van Kampen: proof and identification
Covers the proof and identification of isomorphisms in the Seifert van Kampen theorem.
Related publications (88)
Related people (14)
Related units (1)