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 ${\displaystyle X}$ with continuous dual ${\displaystyle X'}$ the following are equivalent:

• X is barrelled,
• every ${\displaystyle \sigma (X',X)}$-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 ${\displaystyle \beta (X,X')}$,
• every lower semi-continuous semi-norm on ${\displaystyle X}$ is continuous,
• the 0-neighborhood bases in X and the fundamental families of bounded sets in ${\displaystyle E_{\beta }'}$ 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 ${\displaystyle X}$ 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:

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

References

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