Barrelled set explained

In functional analysis, a subset of a topological vector space (TVS) is called a barrel or a barrelled set if it is closed convex balanced and absorbing.

Barrelled sets play an important role in the definitions of several classes of topological vector spaces, such as barrelled spaces.

Definitions

Let

X be a topological vector space (TVS). A subset of X is called a if it is closed convex balanced and absorbing in X. A subset of X is called and a if it absorbs every bounded subset of X. Every bornivorous subset of X is necessarily an absorbing subset of X.

Let

B _{0} \subseteq X be a subset of a topological vector space X. If B _{0} is a balanced absorbing subset of X and if there exists a sequence \left(B _{i\right)} ^{inf ty} ^{i =1} of balanced absorbing subsets of X such that B _{i+1} + B _{i+1} \subseteq B _{i} for all i = 0 , 1 , \ldots, then B _{0} is called a in X, where moreover, B _{0} is said to be a(n):

• if in addition every

B _{i} is a closed and bornivorous subset of X for every i \geq 0 .

• if in addition every

B _{i} is a closed subset of X for every i \geq 0 .

• if in addition every

B _{i} is a closed and bornivorous subset of X for every i \geq 0 .

In this case,

\left(B _{i\right)} ^{inf ty} ^{i =1} is called a for B _{0.}

Properties

Note that every bornivorous ultrabarrel is an ultrabarrel and that every bornivorous suprabarrel is a suprabarrel.

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.

Bibliography

• Book: Hogbe-Nlend, Henri. Bornologies and functional analysis. North-Holland Publishing Co.. Amsterdam. 1977. xii+144. 0-7204-0712-5. 0500064.
  • • Book: H.H. Schaefer. Topological Vector Spaces. Springer-Verlag. GTM. 3. 1970. 0-387-05380-8.
• Book: 9783540115656. Counterexamples in Topological Vector Spaces. Khaleelulla. S.M.. 1982. Springer-Verlag. Berlin Heidelberg. GTM. 936 . 29–33, 49, 104.
• Book: 9780821807804. The Convenient Setting of Global Analysis. Kriegl. Andreas. 1997. American Mathematical Society. Michor. Peter W.. Mathematical Surveys and Monographs.