Weak Topology - Operator Topologies

Operator Topologies

If X and Y are topological vector spaces, the space L(X,Y) of continuous linear operators f:X → Y may carry a variety of different possible topologies. The naming of such topologies depends on the kind of topology one is using on the target space Y to define operator convergence (Yosida 1980, IV.7 Topologies of linear maps). There are, in general, a vast array of possible operator topologies on L(X,Y), whose naming is not entirely intuitive.

For example, the strong operator topology on L(X,Y) is the topology of pointwise convergence. For instance, if Y is a normed space, then this topology is defined by the seminorms indexed by x∈X:

More generally, if a family of seminorms Q defines the topology on Y, then the seminorms p_q,x on L(X,Y) defining the strong topology are given by

indexed by q∈Q and x∈X.

In particular, see the weak operator topology and weak* operator topology.