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...

 and related areas of 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...

, barrelled spaces are Hausdorff topological vector spaces for which every barrelled set in the space is a neighbourhood for the zero vector. A barrelled set or a barrel in a topological vector space is a set which is convex

Convex set

In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...

, balanced

Balanced set

In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space is a set S so that for all scalars α with |α| ≤ 1\alpha S \subseteq S...

, absorbing

Absorbing set

In functional analysis and related areas of mathematics an absorbing set in a vector space is a set S which can be inflated to include any element of the vector space...

 and closed

Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...

. Barrelled spaces are studied because a form of the Banach–Steinhaus theorem still holds for them.

History

Barreled spaces were introduced by .

Examples

• In a semi normed vector space the unit ball is a barrel.
• Every locally convex topological vector space
  Locally convex topological vector space
  In functional analysis and related areas of mathematics, locally convex topological vector spaces or locally convex spaces are examples of topological vector spaces which generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of ...
  
   has a neighbourhood basis consisting of barrelled sets.
• 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...
  
  s, and in particular Banach space
  Banach space
  In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
  
  s, are barrelled, but generally a normed vector space
  Normed vector space
  In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following properties of "vector length" are crucial....
  
   is not barrelled.
• Montel space
  Montel space
  In functional analysis and related areas of mathematics a Montel space, named after Paul Montel, is any topological vector space in which an analog of Montel's theorem holds. Specifically, a Montel space is a barrelled topological vector space where every closed and bounded set is compact...
  
  s are barrelled. Consequently, strong duals of Montel spaces are barreled (since they are Montel spaces).
• locally convex spaces which are Baire space
  Baire space
  In mathematics, a Baire space is a topological space which, intuitively speaking, is very large and has "enough" points for certain limit processes. It is named in honor of René-Louis Baire who introduced the concept.- Motivation :...
  
  s are barrelled.
• a separated, complete
  Complete space
  In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....
  
   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:...
  
   is barrelled.

Properties

• A locally convex space  with continuous dual is barrelled if and only if it carries the strong topology
  Strong topology (polar topology)
  In functional analysis and related areas of mathematics the strong topology is the finest polar topology, the topology with the most open sets, on a dual pair...
  
  .