# Barrelled space

In functional analysis and related areas of mathematics, **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, balanced, absorbing and closed. Barrelled spaces are studied because a form of the Banach–Steinhaus theorem still holds for them.

## History

Barrelled spaces were introduced by Template:Harvs.

## Examples

- In a semi normed vector space the closed unit ball is a barrel.
- Every locally convex topological vector space has a neighbourhood basis consisting of barrelled sets.
- Fréchet spaces, and in particular Banach spaces, are barrelled, but generally a normed vector space is
*not*barrelled. - Montel spaces are barrelled. Consequently, strong duals of Montel spaces are barrelled (since they are Montel spaces).
- locally convex spaces which are Baire spaces are barrelled.

## Properties

For a Hausdorff locally convex space with continuous dual the following are equivalent:

*X*is barrelled,- every -bounded subset of the continuous dual space
*X'*is equicontinuous (this provides a partial converse to the Banach-Steinhaus theorem),^{[1]} - for all subsets
*A*of the continuous dual space*X'*, the following properties are equivalent:*A*is^{[1]}- equicontinuous,
- relatively weakly compact,
- strongly bounded,
- weakly bounded,

*X*carries the strong topology ,- every lower semi-continuous semi-norm on is continuous,
- the 0-neighborhood bases in
*X*and the fundamental families of bounded sets in correspond to each other by polarity.^{[1]}

In addition,

- Every sequentially complete quasibarrelled space is barrelled.
- A barrelled space need not be Montel, complete, metrizable, unordered Baire-like, nor the inductive limit of Banach spaces.

## Quasi-barrelled spaces

A topological vector space for which every barrelled bornivorous set in the space is a neighbourhood of is called a quasi-barrelled space, where a set is bornivorous if it absorbs all bounded subsets of . Every barrelled space is quasi-barrelled.

For a locally convex space with continuous dual the following are equivalent:

- is quasi-barrelled,
- every bounded lower semi-continuous semi-norm on is continuous,
- every -bounded subset of the continuous dual space is equicontinuous.

## References

- ↑
^{1.0}^{1.1}^{1.2}Schaefer (1999) p. 127, 141, Treves (1995) p. 350

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