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verification. (June 2020) |
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.
Let be a topological vector space (TVS). A subset of is called a barrel if it is closed convex balanced and absorbing in A subset of is called bornivorous [1] and a bornivore if it absorbs every bounded subset of Every bornivorous subset of is necessarily an absorbing subset of
Let be a subset of a topological vector space If is a balanced absorbing subset of and if there exists a sequence of balanced absorbing subsets of such that for all then is called a suprabarrel [2] in where moreover, is said to be a(n):
In this case, is called a defining sequence for [2]
Note that every bornivorous ultrabarrel is an ultrabarrel and that every bornivorous suprabarrel is a suprabarrel.
This article needs additional citations for
verification. (June 2020) |
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.
Let be a topological vector space (TVS). A subset of is called a barrel if it is closed convex balanced and absorbing in A subset of is called bornivorous [1] and a bornivore if it absorbs every bounded subset of Every bornivorous subset of is necessarily an absorbing subset of
Let be a subset of a topological vector space If is a balanced absorbing subset of and if there exists a sequence of balanced absorbing subsets of such that for all then is called a suprabarrel [2] in where moreover, is said to be a(n):
In this case, is called a defining sequence for [2]
Note that every bornivorous ultrabarrel is an ultrabarrel and that every bornivorous suprabarrel is a suprabarrel.