Absorbing amount

In mathematics, an absorbing set denotes a subset of a vector space which can be clearly enlarged with scalars in such a way that at some point every point is contained in it and this no longer leaves the set when it is further enlarged.

Absorbing quantities occur, for example, in the context of locally convex spaces and Minkowski functionals .

definition

Let be a - vector space (mostly or ) as well . ${\displaystyle V}$ ${\displaystyle \mathbb {K} }$ ${\displaystyle \mathbb {K} =\mathbb {R} }$ ${\displaystyle \mathbb {K} =\mathbb {C} }$ ${\displaystyle T\subset V}$

Then the set is said to be absorbing if there is a positive real number for each such that ${\displaystyle T}$ ${\displaystyle x\in V}$ ${\displaystyle r>0}$

${\displaystyle \alpha x\in T}$

for everyone with . ${\displaystyle \alpha }$ ${\displaystyle |\alpha |<r}$

The following definition is equivalent to this: for all there is a real one such that ${\displaystyle x\in V}$ ${\displaystyle r>0}$

${\displaystyle x\in \alpha T}$

for everyone with . The set is thus increased by until it absorbs every element of the vector space . ${\displaystyle \alpha }$ ${\displaystyle |\alpha |>r}$ ${\displaystyle T}$ ${\displaystyle \alpha }$

comment

At first glance, this second formulation seems more natural. The former definition is preferred, however, because it translates naturally to the definition of a bounded set of a topological module (namely, a set that is absorbed by any null neighborhood). Because of the possible existence of zero divisors and the possible non-existence of restricted zero environments, a definition in the sense of the second formulation does not make sense in this case.

example

In a topological vector space (e.g. in a normalized space ) every null neighborhood is absorbent, because if a vector is in , then is , i. H. for sufficiently large . ${\displaystyle U}$ ${\displaystyle x}$ ${\displaystyle V}$ ${\displaystyle \lim _{n\rightarrow \infty }{\frac {1}{n}}x=0}$ ${\displaystyle {\frac {1}{n}}x\in U}$ ${\displaystyle n}$

Simple consequences

Since the requirement is positive, it must contain the zero vector . ${\displaystyle r}$ ${\displaystyle T}$

Furthermore, for each absorbent amount is always

${\displaystyle V=\bigcup _{r>0}rT}$

and the Minkowski functional

${\displaystyle x\mapsto \inf\{\lambda \mid \lambda \geq 0,x\in \lambda T\}}$

is finite. Both properties are partly used for definition.

Web links

• Eric W. Weisstein : Absorbing Set . In: MathWorld (English).

literature

• Dirk Werner : Functional Analysis . 7th, corrected and enlarged edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21016-7 , doi : 10.1007 / 978-3-642-21017-4 .