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Polar topology : ウィキペディア英語版
Polar topology
In functional analysis and related areas of mathematics a polar topology, topology of \mathcal-convergence or topology of uniform convergence on the sets of \mathcal is a method to define locally convex topologies on the vector spaces of a dual pair.
== Definitions ==

Let (X,Y,\langle , \rangle) be a dual pair of vector spaces X and Y over the field \mathbb, either the real or complex numbers.
A set A\subseteq X is said to be ''bounded'' in X with respect to Y, if for each element y\in Y the set of values \ is bounded:
:
\forall y\in Y\qquad \sup_|\langle x,y\rangle|<\infty.

This condition is equivalent to the requirement that the polar A^\circ of the set A in Y
:
A^\circ=\

is an absorbent set in Y, i.e.
:
\bigcup_ be a family of bounded sets in X (with respect to Y) with the following properties:
* each point x of X belongs to some set A\in
:
\forall x\in X\qquad \exists A\in \qquad x\in A,

* each two sets A\in and B\in are contained in some set C\in:
:
\forall A,B\in \qquad \exists C\in \qquad A\cup B\subseteq C,

* is closed under the operation of multiplication by scalars:
:
\forall A\in \qquad \forall\lambda\in\qquad \lambda\cdot A\in .

Then the seminorms of the form
:
\|y\|_A=\sup_|\langle x,y\rangle|,\qquad A\in,

define a Hausdorff locally convex topology on Y which is called the polar topology on Y generated by the family of sets . The sets
:
U_=\,\qquad B\in ,

form a local base of this topology. A net of elements y_i\in Y tends to an element y\in Y in this topology if and only if
:
\forall A\in\qquad \|y_i-y\|_A = \sup_ |\langle x,y_i\rangle-\langle x,y\rangle|\underset0.

Because of this the polar topology is often called the topology of uniform convergence on the sets of \mathcal. The semi norm \|y\|_A is the gauge of the polar set A^\circ.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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