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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

Let $(X,Y,⟨,⟩)$ be a dual pair $(X,Y,⟨,⟩)$ 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 $\{⟨x,y⟩;x\in A\}$ is bounded in $\mathbb{𝔽}$:

$\mathrm{\forall }y\in Y{\sup }_{x\in A}|⟨x,y⟩|<\mathrm{\infty }.$

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

$A^{\circ }=\{y\in Y:{\sup }_{x\in A}|⟨x,y⟩|\le 1\}$

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 $A\in \mathcal{𝒜}$

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

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

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

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

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

Then the seminorms of the form

$\Vert y{\Vert }_A={\sup }_{x\in A}|⟨x,y⟩|,A\in \mathcal{𝒜},$

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

$U_B=\{x\in V:\Vert \phi {\Vert }_B<1\},B\in \mathcal{ℬ},$

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

$\mathrm{\forall }A\in \mathcal{𝒜}\Vert y_i-y{\Vert }_A={\sup }_{x\in A}|⟨x,y_i⟩-⟨x,y⟩|\underset{i\to \mathrm{\infty }}{⟶}0.$

Because of this the polar topology is often called the topology of uniform convergence on the sets of $\mathcal{𝒜}$. The semi norm $\Vert y{\Vert }_A$ 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 $A\subseteq X^{\prime }$ in the dual space $X^{\prime }$.

See also

• Topology consistent with the duality.

Notes

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References

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534