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
Bornological space
Summary
In mathematics, particularly in functional analysis, a bornological space is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that a topological space possesses the minimum amount of structure needed to address questions of continuity.
Bornological spaces are distinguished by the property that a linear map from a bornological space into any locally convex spaces is continuous if and only if it is a bounded linear operator.
Bornological spaces were first studied by George Mackey. The name was coined by Bourbaki after borné, the French word for "bounded".
Bornology
A on a set is a collection of subsets of that satisfy all the following conditions:
covers that is, ;
is stable under inclusions; that is, if and then ;
is stable under finite unions; that is, if then ;
Elements of the collection are called or simply if is understood.
The pair is called a or a .
A or of a bornology is a subset of such that each element of is a subset of some element of Given a collection of subsets of the smallest bornology containing is called the
If and are bornological sets then their on is the bornology having as a base the collection of all sets of the form where and
A subset of is bounded in the product bornology if and only if its image under the canonical projections onto and are both bounded.
If and are bornological sets then a function is said to be a or a (with respect to these bornologies) if it maps -bounded subsets of to -bounded subsets of that is, if
If in addition is a bijection and is also bounded then is called a .
Vector bornology
Let be a vector space over a field where has a bornology
A bornology on is called a if it is stable under vector addition, scalar multiplication, and the formation of balanced hulls (i.e. if the sum of two bounded sets is bounded, etc.).
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.