Operator topologies

In the mathematical field of functional analysis there are several standard topologies which are given to the algebra B(X) of bounded linear operators on a Banach space X.

The topologies listed below are all locally convex, which implies that they are defined by a family of seminorms.

In analysis, a topology is called strong if it has many open sets and weak if it has few open sets, so that the corresponding modes of convergence are, respectively, strong and weak.

(In topology proper, these terms can suggest the opposite meaning, so strong and weak are replaced with, respectively, fine and coarse.)

The diagram on the right is a summary of the relations, with the arrows pointing from strong to weak.

On norm bounded sets of B(H), the weak (operator) and ultraweak topologies coincide.

The norm topology is metrizable and the others are not; in fact they fail to be first-countable.

However, when H is separable, all the topologies above are metrizable when restricted to the unit ball (or to any norm-bounded subset).

For example, the dual space of B(H) in the weak or strong operator topology is too small to have much analytic content.

The adjoint map is not continuous in the strong operator and ultrastrong topologies, while the strong* and ultrastrong* topologies are modifications so that the adjoint becomes continuous.