# 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.

## Properties

For a Hausdorff locally convex space ${\displaystyle X}$ with continuous dual ${\displaystyle X'}$ the following are equivalent:

• 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 ${\displaystyle X}$ for which every barrelled bornivorous set in the space is a neighbourhood of ${\displaystyle 0}$ is called a quasi-barrelled space, where a set is bornivorous if it absorbs all bounded subsets of ${\displaystyle X}$. Every barrelled space is quasi-barrelled.

For a locally convex space ${\displaystyle X}$ with continuous dual ${\displaystyle X'}$ the following are equivalent:

## References

1. Schaefer (1999) p. 127, 141, Treves (1995) p. 350
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