Given a Hilbert space 
, the 
-strong operator topology, also called the ultrastrong
operator topology, is the locally convex topology on
the algebra 
of bounded operators
on 
generated by the seminorms
![p_((x_n))(S)=[sum_(n=1)^infty||Sx_n||^2]^(1/2),](/images/equations/Sigma-StrongOperatorTopology/NumberedEquation1.svg) |
where 
ranges over all sequences in 
such that 
.
A net 
in 
converges 
-strongly to 
precisely when 
tends to 0 for every such square-summable sequence 
. The 
-strong operator topology is stronger than the strong operator
topology; the two topologies agree on norm-bounded subsets of 
.
The 
-strong
topology is important for a number of reasons, not the least of which is its application
to the study of von Neumann algebras. What's
more, the notion of the 
-strong topology is merely one in a larger hierarchical
class of operator topologies on 
which includes the 
-weak topology, the 
-strong* topology, etc.; this hierarchy is the focus of
considerable study in its own right.
-Algebras and von Neumann Algebras." 2013.