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 if it is closed convex balanced and absorbing in
A subset of is called and a 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 in where moreover, is said to be a(n):
if in addition every is a closed and bornivorous subset of for every
if in addition every is a closed subset of for every
if in addition every is a closed and bornivorous subset of for every
In this case, is called a for
Note that every bornivorous ultrabarrel is an ultrabarrel and that every bornivorous suprabarrel is a suprabarrel.
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.
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In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field with an absolute value function ) is a set such that for all scalars satisfying The balanced hull or balanced envelope of a set is the smallest balanced set containing The balanced core of a set is the largest balanced set contained in Balanced sets are ubiquitous in functional analysis because every neighborhood of the origin in every topological vector space (TVS) contains a balanced neig
In functional analysis and related areas of mathematics an absorbing set in a vector space is a set which can be "inflated" or "scaled up" to eventually always include any given point of the vector space. Alternative terms are radial or absorbent set. Every neighborhood of the origin in every topological vector space is an absorbing subset.
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is also a topological space with the property that the vector space operations (vector addition and scalar multiplication) are also continuous functions. Such a topology is called a and every topological vector space has a uniform topological structure, allowing a notion of uniform convergence and completeness.