Editing Operator topologies (section)

== Relations between the topologies ==

The continuous linear functionals on {{math|B(''H'')}} for the weak, strong, and strong<sup>*</sup> (operator) topologies are the same, and are the finite linear combinations of the linear functionals
(x''h''<sub>1</sub>, ''h''<sub>2</sub>) for {{math|''h''<sub>1</sub>, ''h''<sub>2</sub> ∈ ''H''}}. 
The continuous linear functionals on {{math|B(''H'')}} for the ultraweak, ultrastrong, ultrastrong<sup>*</sup> and Arens-Mackey topologies are the same, and are the elements of the predual {{math|B(''H'')<sub>*</sub>}}. 

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 {{math|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 {{math|B(''H'')}}. 
Same is true for the Arens-Mackey topology, the ultrastrong<sup>*</sup>, and the strong<sup>*</sup> topology.

In locally convex spaces, closure of convex sets can be characterized by the continuous linear functionals. Therefore, for a [[convex set|convex]] subset {{mvar|K}} of {{math|B(''H'')}}, the conditions that {{mvar|K}} be closed in the ultrastrong<sup>*</sup>, ultrastrong, and ultraweak topologies are all equivalent and are also equivalent to the conditions that
for all {{math|''r'' > 0}}, {{mvar|K}} has closed intersection with the closed ball of radius {{mvar|r}} in the strong<sup>*</sup>, 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 {{mvar|H}} is separable, all the topologies above are metrizable when restricted to the unit ball (or to any norm-bounded subset).