Editing Operator topologies (section)

== List of topologies on B(''H'') ==

[[Image:Optop.svg|right|thumb|Diagram of relations among topologies on the space {{math|B(''X'')}} of bounded operators]]

There are many topologies that can be defined on {{math|B(''X'')}} besides the ones used above; most are at first only defined when {{math|1=''X'' = ''H''}} is a Hilbert space, even though in many cases there are appropriate generalisations. 
The topologies listed below are all locally convex, which implies that they are defined by a family of [[seminorm]]s. 

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.

If {{mvar|H}} is a Hilbert space, the linear space of [[Hilbert space]] operators {{math|B(''X'')}} has a (unique) [[predual]] <math>B(H)_*</math>,
consisting of the trace class operators, whose dual is {{math|B(''X'')}}. 
The seminorm {{math|''p''<sub>''w''</sub>(''x'')}} for ''w'' positive in the predual is defined to be
{{math|B(''w'', ''x<sup>*</sup>x'')<sup>1/2</sup>}}.

If {{mvar|B}} is a vector space of linear maps on the vector space {{mvar|A}}, then {{math|σ(''A'', ''B'')}} is defined to be the weakest topology on {{mvar|A}} such that all elements of {{mvar|B}} are continuous. 

* The '''[[norm topology]]''' or '''uniform topology''' or '''uniform operator topology''' is defined by the usual norm ||''x''|| on {{math|B(''H'')}}. It is stronger than all the other topologies below. 
* The '''[[Weak topology|weak (Banach space) topology]]''' is {{math|σ(B(''H''), B(''H'')<sup>*</sup>)}}, in other words the weakest topology such that all elements of the dual {{math|B(''H'')<sup>*</sup>}} are continuous. It is the weak topology on the Banach space {{math|B(''H'')}}. It is stronger than the ultraweak and weak operator topologies. (Warning: the weak Banach space topology and the weak operator topology and the ultraweak topology are all sometimes called the weak topology, but they are different.)
* The '''[[Mackey topology]]''' or '''Arens-Mackey topology''' is the strongest locally convex topology on {{math|B(''H'')}} such that the dual is {{math|B(''H'')<sub>*</sub>}}, and is also the uniform convergence topology on {{math|Bσ(B(''H'')<sub>*</sub>}}, {{math|B(''H'')}}-compact convex subsets of {{math|B(''H'')<sub>*</sub>}}. It is stronger than all topologies below. 
* The '''σ-strong-<sup>*</sup> topology''' or '''ultrastrong-<sup>*</sup> topology''' is the weakest topology stronger than the ultrastrong topology such that the adjoint map is continuous. It is defined by the family of seminorms {{math|''p''<sub>''w''</sub>(''x'')}} and {{math|''p''<sub>''w''</sub>(''x''<sup>*</sup>)}} for positive elements {{mvar|w}} of {{math|B(''H'')<sub>*</sub>}}. It is stronger than all topologies below.
*The '''σ-strong topology''' or '''[[ultrastrong topology]]''' or '''strongest topology''' or '''strongest operator topology''' is defined by the family of seminorms {{math|''p''<sub>''w''</sub>(''x'')}} for positive elements {{mvar|w}} of {{math|B(''H'')<sub>*</sub>}}. It is stronger than all the topologies below other than the strong<sup>*</sup> topology. Warning: in spite of the name "strongest topology", it is weaker than the norm topology.)
*The '''σ-weak topology''' or '''ultraweak topology''' or '''[[weak-star operator topology|weak-<sup>*</sup> operator topology]]''' or '''weak-* topology''' or '''weak topology''' or '''{{math|σ(B(''H''), B(''H'')<sub>*</sub>}}) topology''' is defined by the family of seminorms |(''w'', ''x'')| for elements ''w'' of {{math|B(''H'')<sub>*</sub>}}. It is stronger than the weak operator topology. (Warning: the weak Banach space topology and the weak operator topology and the ultraweak topology are all sometimes called the weak topology, but they are different.)
* The '''[[Strong-* operator topology|strong-<sup>*</sup> operator topology]]''' or '''strong-<sup>*</sup> topology''' is defined by the seminorms ||''x''(''h'')|| and ||''x''<sup>*</sup>(''h'')|| for {{math|''h'' ∈ ''H''}}. It is stronger than the strong and weak operator topologies.
* The '''[[strong operator topology]]''' (SOT) or '''strong topology''' is defined by the seminorms ||''x''(''h'')|| for {{math|''h'' ∈ ''H''}}. It is stronger than the weak operator topology.
* The '''[[weak operator topology]]''' (WOT) or '''weak topology''' is defined by the seminorms |(''x''(''h''<sub>1</sub>), ''h''<sub>2</sub>)| for {{math|''h''<sub>1</sub>, ''h''<sub>2</sub> ∈ ''H''}}. (Warning: the weak Banach space topology, the weak operator topology, and the ultraweak topology are all sometimes called the weak topology, but they are different.)