# 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 Bourbaki (1950).

## Properties

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

• X is barrelled,
• every $\sigma(X', X)$-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 $\beta(X, X')$,
• every lower semi-continuous semi-norm on $X$ is continuous,
• the 0-neighborhood bases in X and the fundamental families of bounded sets in $E_{\beta}'$ correspond to each other by polarity.[1]

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

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

• $X$ is quasi-barrelled,
• every bounded lower semi-continuous semi-norm on $X$ is continuous,
• every $\beta(X', X)$-bounded subset of the continuous dual space $X'$ is equicontinuous.

## References

1. ^ a b c Schaefer (1999) p. 127, 141, Treves (1995) p. 350