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California Digital Library higherenglishgra00bainrich (User talk:Fæ/IA books#Fork20) (batch #56512) File usage No pages on the English Wikipedia use this file (pages on other projects are not listed).
It is stronger than the weak operator topology. The weak operator topology (WOT) or weak topology is defined by the seminorms |(x(h 1), h 2)| for h 1, h 2 ∈ H. (Warning: the weak Banach space topology, the weak operator topology, and the ultraweak topology are all sometimes called the weak topology, but they are different.)
Here, "Op x" is the empty operator and "x" is the variable bound by that operator, functioning as the object of the verb "please". Part of the reason to assume the empty operator—variable dependency in such sentences is that they exhibit sensitivity to extraction islands. For example, the following attempt to create a similar example results ...
Operator grammar posits three universal constraints: dependency (certain words depend on the presence of other words to form an utterance), likelihood (some combinations of words and their dependents are more likely than others) and reduction (words in high likelihood combinations can be reduced to shorter forms, and sometimes omitted completely).
In this definition the weak (operator) topology can be replaced by many other common topologies including the strong, ultrastrong or ultraweak operator topologies. The *-algebras of bounded operators that are closed in the norm topology are C*-algebras , so in particular any von Neumann algebra is a C*-algebra.
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 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 ...
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