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).

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, although the space itself need not be a barreled space.
Fréchet spaces, and in particular Banach spaces, are barrelled, but generally a normed vector space is not barrelled. For instance, if L²([0, 1]) is topologized as a subspace of L¹([0, 1]), then it is not barrelled.
Montel spaces are barrelled. Consequently, strong duals of Montel spaces are barrelled (since they are Montel spaces).
Every topological vector space which is of the second category in itself is barrelled.

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]

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 $X$ , where every bornivorous^[2] barrel is a neighbourhood of $0$ , is called a quasi-barrelled space^[3]. 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 2 3 Schaefer (1999) p. 127, 141, Treves (1995) p. 350
↑ A convex balanced set $B$ in a topological vector space $X$ is said to be bornivorous if it absorbs each bounded subset $S\subseteq X$ , i.e. $S\subseteq \lambda \cdot B$ for some $\lambda >0$ .
↑ Jarhow 1981, p. 222.

Bourbaki, Nicolas (1950). "Sur certains espaces vectoriels topologiques". Annales de l'Institut Fourier (in French). 2: 5–16 (1951). MR 0042609.
Robertson, Alex P.; Robertson, Wendy J. (1964). Topological vector spaces. Cambridge Tracts in Mathematics. 53. Cambridge University Press. pp. 65–75.
Schaefer, Helmut H. (1971). Topological vector spaces. GTM. 3. New York: Springer-Verlag. p. 60. ISBN 0-387-98726-6.
S.M. Khaleelulla (1982). Counterexamples in Topological Vector Spaces. GTM. 936. Springer-Verlag. pp. 28–46. ISBN 978-3-540-11565-6.
Jarhow, Hans (1981). Locally convex spaces. Teubner. ISBN 978-3-322-90561-1.

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[Schaefer_(1999)_p._127,_141,_Treves_(1995)_p._350-1] 1 2 3 Schaefer (1999) p. 127, 141, Treves (1995) p. 350

[2] A convex balanced set $B$ in a topological vector space $X$ is said to be bornivorous if it absorbs each bounded subset $S\subseteq X$ , i.e. $S\subseteq \lambda \cdot B$ for some $\lambda >0$ .

[FOOTNOTEJarhow1981222-3] Jarhow 1981, p. 222.

Functional analysis (topics)
TVS types	Banach Banach Lattice Barrelled Bornological Brauner F-space Finite-dimensional Fréchet (tame) Hilbert (pre-Hilbert Polarization identity) LF-space Locally convex (Seminorms/Minkowski functionals) Mackey Montel Nuclear Normed (norm) Quasinormed Reflexive Riesz Smith Stereotype Strictly convex Webbed Topological tensor product (of Hilbert spaces)
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Types of sets	Absolutely convex Absorbing Balanced Bounded Convex Convex cone (subset) Linear cone (subset) Radial Star-shaped Symmetric Zonotope
Subsets / set operations	Algebraic interior (core) Bounding points Convex hull Extreme point Interior Minkowski addition Polar