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Concept
Absorbing set
Summary
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.
Official source
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