Operator topologies

In the mathematical field of functional analysis there are several standard topologies which are given to the algebra B(X) of bounded linear operators on a Banach space X. Let be a sequence of linear operators on the Banach space X. Consider the statement that converges to some operator T on X. This could have several different meanings: If , that is, the operator norm of (the supremum of , where x ranges over the unit ball in X ) 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 x ∈ X we have in the weak topology of X. This means that for all continuous linear functionals F on X. In this case we say that in the weak operator topology. There are many topologies that can be defined on B(X) besides the ones used above; most are at first only defined when 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 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 H is a Hilbert space, the Hilbert space B(X) has a (unique) predual , consisting of the trace class operators, whose dual is B(X). The seminorm pw(x) for w positive in the predual is defined to be B(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). It is stronger than all the other topologies below.