Polar crowd

The polar set or the polar of a set is a mathematical term from the mathematical branch of functional analysis . A set of the dual space is assigned to a set of a vector space and vice versa.

definition

If a normalized space or more generally a locally convex space with dual space and is a subset, then one calls ${\displaystyle E}$ ${\displaystyle E'}$ ${\displaystyle A\subset E}$

${\displaystyle A^{\circ }:=\{y\in E':\sup\{|y(x)|:x\in A\}\leq 1\}\subset E'}$

the polar of . ${\displaystyle A}$

Is , so you bet ${\displaystyle B\subset E'}$

${\displaystyle ^{\circ }B:=\{x\in E:\sup\{|y(x)|:y\in B\}\leq 1\}\subset E}$

and calls this the polar of . Often you can find the spelling for this and accept the ambiguity that goes with it, because according to the above definition, a subset of the bidual space would be . ${\displaystyle B}$ ${\displaystyle B^{\circ }}$ ${\displaystyle B^{\circ }}$ ${\displaystyle E^{''}}$

Examples

• The polar of the unit sphere of a normalized space is the unit sphere of the dual space.
• If a subspace , then the nullator of . ${\displaystyle F\subset E}$ ${\displaystyle F^{\circ }}$ ${\displaystyle F}$

properties

The following applies to quantities : ${\displaystyle A,B,A_{i}\subseteq E}$

• From follows ${\displaystyle A\subseteq B}$ ${\displaystyle B^{\circ }\subseteq A^{\circ }}$
• For all true ${\displaystyle \lambda \neq 0}$ ${\displaystyle (\lambda A)^{\circ }={\frac {1}{\lambda }}A^{\circ }}$
• ${\displaystyle (\bigcup _{i\in I}A_{i})^{\circ }=\bigcap _{i\in I}A_{i}^{\circ }}$for a family of subsets ${\displaystyle (A_{i})_{i\in I}}$
• ${\displaystyle A^{\circ }}$is absolutely convex and weak - * - closed .

Applications

The most important theorems about polar sets are:

• Bipolar theorem  : If , then is the absolutely convex, weak - * - closed envelope of . ${\displaystyle A\subset E}$ ${\displaystyle ^{\circ }(A^{\circ })}$ ${\displaystyle A}$

So if absolutely convex and weakly - * - closed, then we have . This can be seen as a simple consequence of the separation theorem. ${\displaystyle A}$ ${\displaystyle ^{\circ }(A^{\circ })=A}$

• Banach-Alaoglu theorem : The polar of a null neighborhood is weak - * - compact.

Some locally convex topologies can be described quite easily using polar sets:

• The set of all polars of all finite sets of dual chamber forms a base of neighborhoods of the weak topology on . ${\displaystyle E}$
• The set of all polars of all finite sets of the vector space forms a base of neighborhoods of the weak - * - topology on ${\displaystyle E'}$
• The set of all polars of all absolutely convex, weakly - * - compact subsets of the dual space forms a base of neighborhoods of the Mackey topology on . ${\displaystyle E}$
• The set of all polars of all weak - * - bounded subsets of the dual space forms a base of neighborhoods of the so-called strong topology on . ${\displaystyle E}$

Individual evidence

1. ^ R. Meise, D. Vogt: Introduction to functional analysis , Vieweg, 1992 ISBN 3-528-07262-8 , §6, §22
2. ↑ H. Heuser: functional analysis , Teubner-Verlag (2006), ISBN 3-8351-0026-2 , sentence 67.2
3. ↑ R. Meise, D. Vogt: Introduction to functional analysis , Vieweg, 1992 ISBN 3-528-07262-8 , sentence 23.5
4. ^ R. Meise, D. Vogt: Introduction to Functional Analysis , Vieweg, 1992 ISBN 3-528-07262-8 , § 23