Topologies on the set of operators on a Hilbert space

In mathematics, the requirements of functional analysis mean there are several standard topologies which are given to the set of bounded linear operators on a Hilbert space.

• norm topology
• ultrastrong topology
• strong operator topology (SOT)
• weak operator topology (WOT)
• weak-star operator topology
• ultraweak topology

The norm topology is stronger than the strong operator topology which is stronger than the weak operator topology.

The norm topology is stronger than the weak-star topology which is stronger than the weak operator topology.

The closures of a convex set in the strong and the weak operator topologies coincide.

The weak operator topology and the weak-star topology agree on norm-bounded sets.

See also:

• topology
• Hilbert space
• bounded operator