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.
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 $X$ 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 $X$ 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 2 3 Schaefer (1999) p. 127, 141, Treves (1995) p. 350

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.

Functional analysis

Set / subset types	Absolutely convex Absorbing Balanced Bounded Convex Radial Star-shaped Symmetric Linear cone (subset) Convex cone (subset)

TVS types	Banach Barrelled Bornological Brauner F-space Finite-dimensional Fréchet (tame) Hilbert LF-space Locally convex Mackey Montel Nuclear Normed (norm) Quasinormed Reflexive Riesz Smith Stereotype Webbed Topological tensor product (of Hilbert spaces)

Mapping topologies	Dual Dual space Operator Ultraweak Weak (operator) Mackey Strong (polar operator) Ultrastrong Uniform convergence

Linear operators	Adjoint Bilinear (form) Bounded / Unbounded Closed Continuous / Discontinuous Compact Fredholm Hilbert–Schmidt Functionals (positive) Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary

Set operations	Algebraic interior (core) Interior Minkowski addition Polar

Operator theory	Banach algebras C-algebras Spectrum (C-algebra radius) Spectral theory (of ODEs Spectral theorem) Polar decomposition Singular value decomposition

Theorems	Banach–Alaoglu Banach–Saks Banach–Mazur Bessel's inequality Cauchy–Schwarz inequality Closed range Closed graph Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Goldstine Hahn–Banach (hyperplane separation) Kakutani fixed-point Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Open mapping Parseval's identity Schauder fixed-point

Analysis	Bochner space Differentiation in Fréchet spaces Derivatives (Fréchet Gâteaux functional holomorphic) Integrals (Bochner Dunford Gelfand–Pettis regulated weak) Functional calculus (Borel continuous holomorphic) Inverse function theorem (Nash–Moser theorem) Vector measure Weakly measurable function

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