In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on a Hilbert space H induced by the seminorms of the form
, as x varies in H. Equivalently, it is the coarsest topology such that, for each fixed x in H, the evaluation map
(taking values in H) is continuous in T. The equivalence of these two definitions can be seen by observing that a subbase for both topologies is given by the sets
(where T0 is any bounded operator on H, x is any vector and ε is any positive real number).
It can be viewed as more natural, too, since it is simply the topology of pointwise convergence.
The linear functionals on the set of bounded operators on a Hilbert space that are continuous in the SOT are precisely those continuous in the weak operator topology (WOT).
This language translates into convergence properties of Hilbert space operators.