Polar topology

Let ${\displaystyle (X,Y,\langle ,\rangle )}$ be a dual pair of vector spaces ${\displaystyle X}$ and ${\displaystyle Y}$ over the field ${\displaystyle \mathbb {F} }$, either the real or complex numbers.

A set ${\displaystyle A\subseteq X}$ is said to be bounded in ${\displaystyle X}$ with respect to ${\displaystyle Y}$, if for each element ${\displaystyle y\in Y}$ the set of values ${\displaystyle \{\langle x,y\rangle ;x\in A\}}$ is bounded:

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

This condition is equivalent to the requirement that the polar ${\displaystyle A^{\circ }}$ of the set ${\displaystyle A}$ in ${\displaystyle Y}$

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

is an absorbent set in ${\displaystyle Y}$, i.e.

${\displaystyle \bigcup _{\lambda \in {\mathbb {F} }}\lambda \cdot A^{\circ }=Y.}$

Let now ${\displaystyle {\mathcal {A}}}$ be a family of bounded sets in ${\displaystyle X}$ (with respect to ${\displaystyle Y}$) with the following properties:

• each point ${\displaystyle x}$ of ${\displaystyle X}$ belongs to some set ${\displaystyle A\in {\mathcal {A}}}$

${\displaystyle \forall x\in X\qquad \exists A\in {\mathcal {A}}\qquad x\in A,}$

• each two sets ${\displaystyle A\in {\mathcal {A}}}$ and ${\displaystyle B\in {\mathcal {A}}}$ are contained in some set ${\displaystyle C\in {\mathcal {A}}}$:

${\displaystyle \forall A,B\in {\mathcal {A}}\qquad \exists C\in {\mathcal {A}}\qquad A\cup B\subseteq C,}$

• ${\displaystyle {\mathcal {A}}}$ is closed under the operation of multiplication by scalars:

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

Then the seminorms of the form

${\displaystyle \|y\|_{A}=\sup _{x\in A}|\langle x,y\rangle |,\qquad A\in {\mathcal {A}},}$

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

${\displaystyle U_{A}=\{\varphi \in Y:\quad \|\varphi \|_{A}<1\},\qquad A\in {\mathcal {A}},}$

form a local base of this topology. A net of elements ${\displaystyle y_{i}\in Y}$ tends to an element ${\displaystyle y\in Y}$ in this topology if and only if

${\displaystyle \forall A\in {\mathcal {A}}\qquad \|y_{i}-y\|_{A}=\sup _{x\in A}|\langle x,y_{i}\rangle -\langle x,y\rangle |{\underset {i\to \infty }{\longrightarrow }}0.}$

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

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

• Topologies on spaces of linear maps
• Topology consistent with the duality.

• 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.