In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, particularly in functional analysis

Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

, 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

Bounded set

In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. Conversely, a set which is not bounded is called unbounded...

 and functions

Bounded function

In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. In other words, there exists a real number M...

, in the same way that a topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

 possesses the minimum amount of structure needed to address questions of continuity

Continuous function

In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

.

Bornological sets

Let X be any set. A bornology on X is a collection B of subsets of X such that

• B is a covering of X, i.e.

• B is stable under inclusions, i.e. if B ∈ B and A ⊆ B, then A ∈ B;
• B is stable under finite unions, i.e. if B₁, ..., B_n ∈ B, then

Elements of the collection B are called bounded sets and the pair (X, B) is called a bornological set.

Examples

• For any set X, the discrete topology of X is a Bornology.
• For any set X, the set of finite (or countably infinite) subsets of X is a Bornology.
• For any topological space X, the set of subsets of X with compact
  Compact space
  In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
  
   closure
  Closure (topology)
  In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are "near" S. A point which is in the closure of S is a point of closure of S...
  
   is a Bornology.

Bornological spaces in functional analysis

In functional analysis, a bornological space is a locally convex space X such that every semi-norm on X which is bounded on all bounded subsets of X is continuous, where a subset A of X is bounded whenever all continuous semi-norms on X are bounded on A.

Equivalently, a locally convex space X is bornological if and only if the continuous linear operator

Continuous linear operator

In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces....

s on X to any locally convex space Y are exactly the bounded linear operators from X to Y.

This gives a connection to the above definition of a bornology. Every topological vector space X gives a bornology on X by defining a subset to be bounded iff for all open sets containing zero there exists a with .
A locally convex X is bornological iff its topology can be recovered from its bornology in a natural way.

For example, any metrisable locally convex space is bornological. In particular, any Fréchet space

Fréchet space

In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces...

 is bornological.

Given a bornological space X with continuous dual X′, then the topology of X coincides with the Mackey topology

Mackey topology

In functional analysis and related areas of mathematics, the Mackey topology, named after George Mackey, is the finest topology for a topological vector space which still preserves the continuous dual...

 τ(X,X′). In particular, bornological spaces are Mackey space

Mackey space

In mathematics, particularly in functional analysis, a Mackey space is a locally convex space X such that the topology of X coincides with the Mackey topology τ.-Properties:...

s.