Polar topology

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In functional analysis and related areas of mathematics a polar topology, topology of \mathcal{A}-convergence or topology of uniform convergence on the sets of \mathcal{A} is a method to define locally convex topologies on the vector spaces of a dual pair.

Definitions[edit]

Let (X,Y,\langle , \rangle) be a dual pair (X,Y,\langle , \rangle) of vector spaces X and Y over the (same) field \mathbb{F} of 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 \{\langle x,y\rangle; x\in A\} is bounded in \mathbb F:


\forall y\in Y\qquad \sup_{x\in A}|\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=\{ y\in Y:\quad \sup_{x\in A}|\langle x,y\rangle|\le 1\}

is an absorbent set in Y, i.e.


\bigcup_{\lambda\in{\mathbb F}}\lambda\cdot A^\circ=Y.

Let now \mathcal{A} 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{\mathcal A}

    \forall x\in X\qquad \exists A\in {\mathcal A}\qquad x\in A,
  • each two sets A\in{\mathcal A} and B\in{\mathcal A} are contained in some set C\in{\mathcal A}:

    \forall A,B\in {\mathcal A}\qquad \exists C\in {\mathcal A}\qquad A\cup B\subseteq C,
  • {\mathcal A} is closed under the operation of multiplication by scalars:

    \forall A\in {\mathcal A}\qquad \forall\lambda\in{\mathbb F}\qquad \lambda\cdot A\in {\mathcal A}.

Then the seminorms of the form


\|y\|_A=\sup_{x\in A}|\langle x,y\rangle|,\qquad A\in{\mathcal A},

define a Hausdorff locally convex topology on Y which is called the polar topology[1] on Y generated by the family of sets {\mathcal A}. The sets


    U_{B}=\{x\in V:\quad \|\varphi\|_B<1\},\qquad B\in {\mathcal B},

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{\mathcal A}\qquad  \|y_i-y\|_A = \sup_{x\in A} |\langle x,y_i\rangle-\langle x,y\rangle|\underset{i\to\infty}{\longrightarrow}0.

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

Examples[edit]

  • if \mathcal A is the family of all bounded sets in X then the polar topology on Y coincides with the strong topology,
  • if \mathcal A is the family of all finite sets in X then the polar topology on Y coincides with the weak topology,
  • the topology of an arbitrary locally convex space X can be described as the polar topology defined on X by the family \mathcal A of all equicontinuous sets A\subseteq X' in the dual space X'.[2]

See also[edit]

Notes[edit]

  1. ^ A.P.Robertson, W.Robertson (1964, III.2)
  2. ^ In other words, A\in{\mathcal A} iff A\subseteq X' and there is a neighbourhood of zero U\subseteq X such that \sup_{x\in U, f\in A}|f(x)|<\infty

References[edit]

  • Robertson, A.P.; Robertson, W. (1964). Topological vector spaces. Cambridge University Press. 
  • Schaefer, Helmuth H. (1966). Topological vector spaces. New York: The MacMillan Company. ISBN 0-387-98726-6.