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The σ-strong topology or ultrastrong topology or strongest topology or strongest operator topology is defined by the family of seminorms p w (x) for positive elements w of B(H) *. It is stronger than all the topologies below other than the strong * topology.
The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that, see List of general topology topics and Topological property.
The predual of B(H) is the trace class operators C 1 (H), and it generates the w*-topology on B(H), called the weak-star operator topology or σ-weak topology. The weak-operator and σ-weak topologies agree on norm-bounded sets in B(H). A net {T α} ⊂ B(H) converges to T in WOT if and only Tr(T α F) converges to Tr(TF) for all finite-rank ...
The SOT topology also provides the framework for the measurable functional calculus, just as the norm topology does for the continuous functional calculus. The linear functionals on the set of bounded operators on a Hilbert space that are continuous in the SOT are precisely those continuous in the weak operator topology (WOT).
The ultraweak topology is similar to the weak operator topology. For example, on any norm-bounded set the weak operator and ultraweak topologies are the same, and in particular, the unit ball is compact in both topologies. The ultraweak topology is stronger than the weak operator topology.
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators , and consideration may be given to nonlinear operators .
Absolutely closed See H-closed Accessible See . Accumulation point See limit point. Alexandrov topology The topology of a space X is an Alexandrov topology (or is finitely generated) if arbitrary intersections of open sets in X are open, or equivalently, if arbitrary unions of closed sets are closed, or, again equivalently, if the open sets are the upper sets of a poset.
Both the weak topology and the weak* topology are special cases of a more general construction for pairings, which we now describe.The benefit of this more general construction is that any definition or result proved for it applies to both the weak topology and the weak* topology, thereby making redundant the need for many definitions, theorem statements, and proofs.