Introduction
Let {Tn} be a sequence of linear operators on the Hilbert space H. Consider the statement that Tn converges to some operator T in H. This could have several different meanings:
- If, that is, the operator norm of Tn - T (the supremum of, where x ranges over the unit ball in H) converges to 0, we say that in the uniform operator topology.
- If for all x in H, then we say in the strong operator topology.
- Finally, suppose in the weak topology of H. This means that for all linear functionals F on H. In this case we say that in the weak operator topology.
All of these notions make sense and are useful for a Banach space in place of the Hilbert space H.
Read more about this topic: Operator Topologies
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