What Are the Causes of Bipolar?

Let be any compatible topology on X and AX, then A ⁰⁰ = (A ⁰ ) ⁰ is equal to the convex equilibrium closure of A. This proposition is called the bipolar theorem.

When X = Y is a Hilbert space, the pole A ^{0 of the} subspace A of X = A . And when X is Banach space, Y = X *, and

Time,

As the poles can be operated, this brings great convenience to the study of dual space theory.

Maximum computation is one of the very useful tools in topological linear space theory. ^[1]

Compatible topology is a locally convex topology.

Let (X, Y) be a dual linear space. If is a locally convex topology on X such that the continuous linear functional group (X, ) 'on X about is exactly Y, then is a Compatible topology.

Weak topology a (X, Y) is the weakest compatible topology on X.

Compatible topology is a locally convex topology that is well worth studying in dual linear spaces, and all such topologies need to be characterized.