In mathematics, the requirements of functional analysis mean there are several standard topologies which are given to the algebra B(H) of bounded linear operators on a Hilbert space H. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and For functional analysis as used in psychology see the Functional analysis (psychology article Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Functional analysis (a branch of Mathematics) a bounded linear operator is a Linear transformation L between Normed vector spaces This article assumes some familiarity with Analytic geometry and the concept of a limit.

Introduction

Let {T_n} be a sequence of linear operators on the Hilbert space H. Consider the statement that T_n converges to some operator T in H. This could have several different meanings:

• If , that is, the supremum of T_nx - T x converges to 0, where x ranges over the unit ball in H, we say that in the uniform operator topology. In Mathematics, a unit Sphere is the set of points of Distance 1 from a fixed central point where a generalized concept of distance may be used a closed
• If for all x in H, then we say in the strong operator topology.
• Finally, suppose in the weak topology of H. In Mathematics, weak topology is an alternative term for Initial topology. This means that for all linear functionals F on H. This article deals with Linear maps from a Vector space to its field of scalars These maps may be functionals in the traditional In this case we say that in the weak operator topology.

All of these notions make sense and are useful for a Banach space in place of the Hilbert space H. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis

List of topologies on B(H)

Diagram of relations among topologies on the space B(H) of bounded operators

There are many topologies that can be defined on B(H) besides the ones used above. These topologies are all locally convex, which implies that they are defined by a family of seminorms. In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length

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

The Banach space B(H) has a (unique) predual B(H)_*, consisting of the trace class operators, whose dual is B(H). In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis In Mathematics, the predual of an object D is an object P whose Dual space is D. The seminorm p_w(x) for w positive in the predual is defined to be (w, x^*x)^1/2.

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

• The norm topology or uniform topology or uniform operator topology is defined by the usual norm ||x|| on B(H). In Mathematics, the operator norm is a means to measure the "size" of certain Linear operators Formally it is a norm defined on the space of It is stronger than all the other topologies below.
• The weak (Banach space) topology is σ(B(H), B(H)^*), in other words the weakest topology such that all elements of the dual B(H)^* are continuous. In Mathematics, weak topology is an alternative term for Initial topology. It is the weak topology on the Banach space 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 B(H) such that the dual is B(H)_*, and is also the uniform convergence topology on σ(B(H)_*, B(H)-compact convex subsets of B(H)_*. In Functional analysis and related areas of Mathematics, the Mackey topology, named after George Mackey, is the finest topology for a Topological 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 p_w(x) and p_w(x^*) for positive elements w of B(H)_*. 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 p_w(x) for positive elements w of B(H)_*. In Functional analysis, the ultrastrong topology, or &sigma-strong topology, or strongest topology on the set B(H of Bounded operators 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 σ(B(H), B(H)_*) topology is defined by the family of seminorms |(w, x)| for elements w of B(H)_*. In Functional analysis, a branch of Mathematics, the ultraweak topology, also called the weak-* topology, or weak-* operator topology or 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 h in 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 h in H. In Functional analysis, a branch of Mathematics, the strong operator topology, often abbreviated SOT is the weakest Topology on the set of Bounded 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 h₁ and h₂ in H. In Functional analysis, the weak operator topology, often abbreviated WOT is the weakest Topology on the set of Bounded operators on a Hilbert space (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 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)_*. The continuous linear functions in the norm topology form 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. In Functional analysis and related branches of Mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed 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 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. In Euclidean space, an object is convex if for every pair of points within the object every point on the Straight line segment that joins them is also within the

The norm topology is metrizable and the others are not; in fact they fail to be first-countable. In Topology, a branch of Mathematics, a first-countable space is a Topological space satisfying the "first Axiom of countability " However, when H is separable, all the topologies above are metrizable when restricted to the unit ball (or to any norm-bounded subset).

Which topology should I use?

The most commonly used topologies are the norm, strong, and weak operator topologies. The weak operator topology is useful for compactness arguments, because the unit ball is compact by the Banach-Alaoglu theorem. The norm topology is fundamental because it makes B(H) into a Banach space, but it is too strong for many purposes; for example, B(H) is not separable in this topology. The strong operator topology could be the most commonly used.

The ultraweak and ultrastrong topologies are better-behaved than the weak and strong operator topologies, but their definitions are more complicated, so they are usually not used unless their better properties are really needed. 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. They are not used very often.

The Arens-Mackey topology and the weak Banach space topology are very rarely used.

To summarize, the three essential topologies on B(H) are the norm, ultrastrong, and ultraweak topologies. The weak and strong operator topologies are widely used as convenient approximations to the ultraweak and ultrastrong topologies. The other topologies are relatively obscure.

See also

• Topology
• Hilbert space
• Bounded operator

References

• Functional analysis, by Reed and Simon, ISBN 0-12-585050-6
• Theory of Operator Algebras I, by M. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of This article assumes some familiarity with Analytic geometry and the concept of a limit. In Functional analysis (a branch of Mathematics) a bounded linear operator is a Linear transformation L between Normed vector spaces Takesaki (especially chapter II. 2) ISBN 3-540-42248-X