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 $Y$

$A^{\circ }=\{y\in Y:\quad \sup _{{x\in A}}|\langle x,y\rangle |\leq 1\}$

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.