Absorbing set

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. Notation for scalars Suppose that is a vector space over the field of real numbers or complex numbers and for any let denote the open ball (respectively, the closed ball) of radius in centered at Define the product of a set of scalars with a set of vectors as and define the product of with a single vector as Balanced core and balanced hull A subset of is said to be if for all and all scalars satisfying this condition may be written more succinctly as and it holds if and only if Given a set the smallest balanced set containing denoted by is called the of while the largest balanced set contained within denoted by is called the of These sets are given by the formulas and (these formulas show that the balanced hull and the balanced core always exist and are unique). A set is balanced if and only if it is equal to its balanced hull () or to its balanced core (), in which case all three of these sets are equal: If is any scalar then while if is non-zero or if then also If and are subsets of then is said to if it satisfies any of the following equivalent conditions: Definition: There exists a real such that for every scalar satisfying Or stated more succinctly, for some If the scalar field is then intuitively, " absorbs " means that if is perpetually "scaled up" or "inflated" (referring to as ) then (for all positive sufficiently large), all will contain and similarly, must also eventually contain for all negative sufficiently large in magnitude. This definition depends on the underlying scalar field's canonical norm (that is, on the absolute value ), which thus ties this definition to the usual Euclidean topology on the scalar field. Consequently, the definition of an absorbing set (given below) is also tied to this topology.

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Balanced set

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

Bounded set (topological vector space)

In functional analysis and related areas of mathematics, a set in a topological vector space is called bounded or von Neumann bounded, if every neighborhood of the zero vector can be inflated to include the set. A set that is not bounded is called unbounded. Bounded sets are a natural way to define locally convex polar topologies on the vector spaces in a dual pair, as the polar set of a bounded set is an absolutely convex and absorbing set. The concept was first introduced by John von Neumann and Andrey Kolmogorov in 1935.

Metrizable topological vector space

In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of locally convex metrizable TVS.

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