Topologies on the set of operators on a Hilbert space
In the mathematical field of functional analysis there are several standard topologies which are given to the algebra B(H) of bounded linear operators on a Hilbert space H.
Let {Tn} be a sequence of linear operators on the Hilbert space H. Consider the statement that Tn converges to some operator T in H. This could have several different meanings:
All of these notions make sense and are useful for a Banach space in place of the Hilbert space H.
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 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.
The Banach space B(H) has a (unique) predual B(H)*, consisting of the trace class operators, whose dual is B(H). The seminorm pw(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 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)*.
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.
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