Strong vs Weak Operator Topology

For operators $T_n,T$ on a Hilbert space:

• SOT (strong): $T_n\to T$ if $T_nx\to Tx$ for every vector $x$.
• WOT (weak): $T_n\to T$ if $(T_nx,y)\to (Tx,y)$ for all vectors $x,y$.

Key properties (paper use):

• Continuity of group actions/representations is often stated in WOT.
• On unitary/orthogonal groups, these topologies are commonly used for “weak continuity”.

Example: If $\|T_n-T\|\to0$ (operator norm), then $T_n\to T$ in both SOT and WOT.