Balanced amount

In functional analysis, a balanced set denotes a subset of a vector space , which is characterized by the fact that for each element of the set the negative of this element is also included in the set and the entire connection between these two elements. Many authors also use the terms circular (English circled), disk-shaped or balanced (English).

Balanced sets are used, for example, in the definition of locally convex spaces , where balance is a property of the defining zero neighborhood basis .

definition

A real or complex vector space is given . A quantity is called a balanced quantity if there is scalars with and all also for all . For everyone , the route from to is in . ${\ displaystyle V}$ ${\ displaystyle T \ subset V}$ ${\ displaystyle r}$ ${\ displaystyle | r | \ leq 1}$ ${\ displaystyle x \ in T}$ ${\ displaystyle rx \ in T}$ ${\ displaystyle x \ in T}$ ${\ displaystyle -x}$ ${\ displaystyle x}$ ${\ displaystyle T}$

properties

If it is balanced and not empty, then it must contain the zero vector , because is in , so is . ${\ displaystyle T}$ ${\ displaystyle T}$ ${\ displaystyle x}$ ${\ displaystyle T}$ ${\ displaystyle 0 = 0 \ cdot x \ in T}$

In a topological vector space , every neighborhood of zero also contains a balanced zero neighborhood. Namely , if there is a zero neighborhood, then because of the continuity of the scalar multiplication there is one and a zero neighborhood , so that for all and all in . Then there is a zero balanced environment contained in. ${\ displaystyle U}$ ${\ displaystyle \ varepsilon> 0}$ ${\ displaystyle V}$ ${\ displaystyle rx \ in U}$ ${\ displaystyle | r | <\ varepsilon}$ ${\ displaystyle x}$ ${\ displaystyle V}$ ${\ displaystyle \ textstyle \ bigcup _ {| r | <\ varepsilon} rV}$ ${\ displaystyle U}$

In a topological vector space there is always a zero neighborhood basis made up of balanced sets. Conversely, if one has a system of absorbing and balanced sets with the properties on an algebraic vector space ${\ displaystyle {\ mathcal {U}}}$

The following applies to all ${\ displaystyle U \ in {\ mathcal {U}}, r> 0}$ ${\ displaystyle rU \ in {\ mathcal {U}}}$
${\ displaystyle {\ mathcal {U}}}$ with two quantities each also contains their average,
For each there is a with , ${\ displaystyle U \ in {\ mathcal {U}}}$ ${\ displaystyle V \ in {\ mathcal {U}}}$ ${\ displaystyle V + V \ subset U}$
${\ displaystyle \ bigcap {\ mathcal {U}} = \ {0 \}}$ ,

in this way the vector space becomes a topological vector space with as a zero neighborhood basis. The balance is needed to show the continuity of the scalar multiplication. ${\ displaystyle {\ mathcal {U}}}$

Balanced convex sets are also called absolutely convex . They play an important role in the theory of locally convex spaces .

Web links

VI Sobolev: Balanced Set . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
Eric W. Weisstein : Balanced Set . In: MathWorld (English).

literature

K. Floret, J. Wloka: Introduction to the Theory of Locally Convex Spaces , Lecture Notes in Mathematics 56, 1968
R. Meise, D. Vogt: Introduction to Functional Analysis , Vieweg, 1992 ISBN 3-528-07262-8