Polar topology

is a method to define locally convex topologies on the vector spaces of a pairing.

and defined by In this case, the absolute polar of a subset

The polar is a convex balanced set containing the origin.

[3] Similarly, there are the dual definition of the weak topology on

[3] There is a repeating theme in duality theory, which is that any definition for a pairing

In particular, although this article will only define the general notion of polar topologies on

this article will nevertheless use the dual definition for polar topologies on

Although it is technically incorrect and an abuse of notation, the following convention is nearly ubiquitous: Throughout,

is a directed set with respect to subset inclusion (i.e. if for all

then the defining neighborhood subbasis at the origin can be replaced with without changing the resulting topology.

also satisfy the following additional conditions: Some authors[4] further assume that every

is a TVS topology, it suffices to show that the set

belongs to this continuous dual space if and only if there exists some

are vector spaces over the complex numbers (which implies that

This ultimately stems from the fact that for any complex-valued linear functional

The table below defines many of the most common polar topologies on

The reason why some of the above collections (in the same row) induce the same polar topologies is due to some basic results.

For this reason, it is common to restrict attention to families of bounded subsets of

has compact closure in the topology of uniform convergence on precompact sets.

be the set of all convex balanced weakly compact subsets of

or the topology of uniform convergence on convex balanced weakly compact sets, which is denoted by

Due to the importance of this topology, the continuous dual space of

be the set of all convex balanced weakly compact subsets of

or the topology of uniform convergence on convex balanced weakly compact subsets of

The table below defines many of the most common polar topologies on

was the set of all convex balanced weakly compact equicontinuous subsets of

Importantly, a set of continuous linear functionals

Since a TVS's topology is completely determined by the open neighborhoods of the origin, this means that via operation of taking the polar of a set, the collection of equicontinuous subsets of

is a Hausdorff locally convex topological vector space (TVS) and

The question arises: what are all of the locally convex Hausdorff TVS topologies that can be placed on