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. Definition 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 Preliminaries 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 The (respectively, the ) of a set denoted by (respectively, by ), is defined to be the smallest balanced set containing (respectively, the largest balanced set contained in ). The balanced hull and core always exist and are unique. They are given by the formulas 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 and if is non-zero or if then also One set absorbing another 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. There exists a real such that for every non-zero scalar satisfying Or stated more succinctly, for some Because this union is equal to where is the closed ball