In the mathematical field of functional analysis there are several standard topologies which are given to the algebra of bounded linear operators on a Banach space .
Introduction
Let be a sequence of linear operators on the Banach space . Consider the statement that converges to some operator on . This could have several different meanings: If , that is, the operator norm of (the supremum of , where ranges over the unit ball in ) converges to 0, we say that in the uniform operator topology. If for all , then we say in the strong operator topology. Finally, suppose that for all we have in the weak topology of . This means that for all continuous linear functionals on . In this case we say that in the weak operator topology.
List of topologies on B(H)
There are many topologies that can be defined on besides the ones used above; most are at first only defined when 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 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.
If is a Hilbert space, the Hilbert space has a (unique) predual , consisting of the trace class operators, whose dual is . The seminorm for w positive in the predual is defined to be .
If is a vector space of linear maps on the vector space , then is defined to be the weakest topology on such that all elements of are continuous.
The norm topology or uniform topology or uniform operator topology is defined by the usual norm ||x|| on . It is stronger than all the other topologies below. The weak (Banach space) topology is , in other words the weakest topology such that all elements of the dual are continuous. It is the weak topology on the Banach space . 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 such that the dual is , and is also the uniform convergence topology on , -compact convex subsets of . It is stronger than all topologies below. The σ-strong-* topology or ultrastrong-* 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 and for positive elements of . 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 for positive elements of . It is stronger than all the topologies below other than the strong* 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-* operator topology or weak-* topology or weak topology or ) topology is defined by the family of seminorms |(w, x)| for elements w of . 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 or strong-* topology is defined by the seminorms ||x(h)|| and ||x*(h)|| for . 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 . It is stronger than the weak operator topology. The weak operator topology (WOT) or weak topology is defined by the seminorms |(x(h1), h2)| for . (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.)
Relations between the topologies
The continuous linear functionals on for the weak, strong, and strong* (operator) topologies are the same, and are the finite linear combinations of the linear functionals (xh1, h2) for . The continuous linear functionals on for the ultraweak, ultrastrong, ultrastrong* and Arens-Mackey topologies are the same, and are the elements of the predual .