Operator Topologies - Relations Between The Topologies

Relations Between The Topologies

The continuous linear functionals on B(H) for the weak, strong, and strong* (operator) topologies are the same, and are the finite linear combinations of the linear functionals (xh₁, h₂) for h₁, h₂ in H. The continuous linear functionals on B(H) for the ultraweak, ultrastrong, ultrastrong* and Arens-Mackey topologies are the same, and are the elements of the predual B(H)_*.

By definition, the continuous linear functionals in the norm topology are the same as those in the weak Banach space topology. This dual is a rather large space with many pathological elements.

On norm bounded sets of B(H), the weak (operator) and ultraweak topologies coincide. This can be seen via, for instance, the Banach–Alaoglu theorem. For essentially the same reason, the ultrastrong topology is the same as the strong topology on any (norm) bounded subset of B(H). Same is true for the Arens-Mackey topology, the ultrastrong*, and the strong* topology.

In locally convex spaces, closure of convex sets can be characterized by the continuous linear functionals. Therefore, for a convex subset K of B(H), the conditions that K be closed in the ultrastrong*, ultrastrong, and ultraweak topologies are all equivalent and are also equivalent to the conditions that for all r > 0, K has closed intersection with the closed ball of radius r in the strong*, strong, or weak (operator) topologies.

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).