Polar topology

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

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

$\forall y\in Y\qquad \sup_{x\in A}|\langle x,y\rangle|<\infty.$

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

• 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

• Topology consistent with the duality.

Notes

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

• 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.