Strong operator topology
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} (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 U ( T 0 , x , ϵ ) = { T : ‖ T x − T 0 x ‖ < ϵ } {\displaystyle U(T_{0},x,\epsilon )=\{T:\|Tx-T_{0}x\|<\epsilon \}} (where T0 is any bounded operator on H, x is any vector and ε is any positive real number). In concrete terms, this means that T i → T {\displaystyle T_{i}\to T} in the strong operator topology if and only if ‖ T i x − T x ‖ → 0 {\displaystyle \|T_{i}x-Tx\|\to 0} for each x in H. The SOT is stronger than the weak operator topology and weaker than the norm topology.