Polar set

The polar of the polar curve or a set is a mathematical term from the mathematical branch of functional analysis. In this case, an amount of a vector space is allocated an amount of the dual- chamber and vice versa.

Definition

If a normed space, or more generally a locally convex space with dual space and is a subset of so called

The polar of.

If, as is given to

And calls this the polar of. Often possible to find this the spelling and takes the associated ambiguity in buying, because according to the above definition would be a subset of the Bidualraums.

Examples

• The polar of the unit ball of a normed space is the unit sphere of the dual space.
• Is a subspace, then the Annullator of.

Properties

For quantities apply:

• It follows from
• For all
• For a family of subsets
• Is absolutely convex and weak - * -closed.

Applications

The main theorems on polar quantities are:

• Bipolars: If, as is the absolutely convex, weakly -* closed hull of.

So is absolutely convex and weak - * -closed, so is true. This can be viewed as a simple consequence of the separation theorem.

• Banach - Alaoglu: Polars a neighborhood of zero is weak -* compact.

By means of polar quantities of some locally convex topologies can be fairly simple to describe:

• The set of all polars of all finite sets of the vector space forms a base of neighborhoods of the weak -* topology on
• The set of all polars of all absolutely convex, weakly -* compact subsets of the dual space forms a base of neighborhoods of the Mackey topology.
• The set of all polars of all weak -* bounded subsets of the dual space forms a base of neighborhoods of the so-called strong topology.