“Barrelled space”的意思、由来-开放百科全书

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Barrelled space

释义

History
Examples
Properties
Quasi-barrelled spaces
References

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 {{harvs|last=Bourbaki|authorlink=Nicolas Bourbaki|year=1950|txt}}.

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 with continuous dual the following are equivalent:

X is barrelled,
every -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 ,
every lower semi-continuous semi-norm on is continuous,
the 0-neighborhood bases in X and the fundamental families of bounded sets in 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 , where every bornivorous^[2] barrel is a neighbourhood of , is called a quasi-barrelled space{{sfn|Jarhow|1981|p=222}}. Every barrelled space is quasi-barrelled.

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

is quasi-barrelled,
every bounded lower semi-continuous semi-norm on is continuous,
every -bounded subset of the continuous dual space is equicontinuous.

References

1. ^¹²Schaefer (1999) p. 127, 141, Treves (1995) p. 350
2. ^A convex balanced set

in a topological vector space

is said to be bornivorous if it absorbs each bounded subset

, i.e.

for some

*{{cite journal

| last = Bourbaki | first = Nicolas | authorlink = Nicolas Bourbaki
| journal = Annales de l'Institut Fourier
| language = French
| mr = 0042609
| pages = 5–16 (1951)
| title = Sur certains espaces vectoriels topologiques
| url = http://www.numdam.org/item?id=AIF_1950__2__5_0
| volume = 2
| year = 1950}}

{{cite book |last1=Robertson |first1=Alex P. |first2= Wendy J.|last2=Robertson |title= Topological vector spaces |series=Cambridge Tracts in Mathematics |volume=53 |year=1964 |publisher= Cambridge University Press | pages=65–75}}
{{cite book | last = Schaefer | first = Helmut H. | year = 1971 | title = Topological vector spaces | series=GTM | volume=3 | publisher = Springer-Verlag | location = New York | isbn = 0-387-98726-6 | page=60 }}
{{cite book | author=S.M. Khaleelulla | title=Counterexamples in Topological Vector Spaces | publisher=Springer-Verlag | series=GTM | volume=936 | date=1982 | isbn=978-3-540-11565-6 | pages=28-46 }}
{{cite book |last=Jarhow |first=Hans |title= Locally convex spaces |year=1981 |publisher= Teubner|ISBN=978-3-322-90561-1| ref = harv}}

1 : Topological vector spaces

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