Definition:Absorbent Set

Definition

Let $V$ be a vector space over a field $K$.

Let $W \subseteq V$ be a subset of $V$.

Let $a \in K$.

Let the set $a \cdot W$ be defined as:

$a \cdot W := \left\{{a \cdot y: y \in W} \right\}$

Then $W$ is an absorbent set in $V$ if and only if:

$\ds \bigcup_{a \mathop \in K} a \cdot W = V$

which symbolically can be represented as:

$K \cdot W = V$

Also known as

An absorbent set is also known as an absorbing set or a radial set.

Sources

• 1964: A.P. Robertson and W.J. Robertson: Topological Vector Spaces
• 1966: Helmut H. Schaefer: Topological Vector Spaces
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): absorbing set
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): absorbing set