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 T ↦ ‖ T x ‖ {\displaystyle T\mapsto \|Tx\|}, as x varies in H. Equivalently, it is the coarsest topology such that, for each fixed x in H, the evaluation map T ↦ T x {\displaystyle T\mapsto Tx} 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 U = { T : ‖ T x − T 0 x ‖ < ϵ } {\displaystyle U=\{T:\|Tx-T_{0}x\|<\epsilon \}}. Wikipedia