The strong operator topology on bounded operators on a Hilbert space is the topology of pointwise norm convergence on vectors. A net 
converges strongly to 
when 
for every vector 
. A basic neighborhood tests finitely many vectors with a
positive tolerance. Convergence in operator norm
implies convergence in this topology, but the converse fails in general. For example,
the projections onto the first 
standard basis vectors in
converge strongly to the identity and stay at norm distance 1 from it. Strong-operator paving uses projections whose complements
lie in prescribed neighborhoods of zero in this topology.
Strong Operator Topology
See also
Bounded Operator, Hilbert Space, Operator Norm, Operator Paving, Sigma-Strong Operator TopologyExplore with Wolfram|Alpha
References
Anantharaman, C. and Popa, S. An Introduction to
Factors. https://www.math.ucla.edu/~popa/Books/IIun.pdf.
Cite this as:
Weisstein, Eric W. "Strong Operator Topology." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/StrongOperatorTopology.html