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The definition of weak convergence can be extended to Banach spaces. A sequence of points ( x n ) {\displaystyle (x_{n})} in a Banach space B is said to converge weakly to a point x in B if f ( x n ) → f ( x ) {\displaystyle f(x_{n})\to f(x)} for any bounded linear functional f {\displaystyle f} defined on B {\displaystyle B} , that is, for ...
In mathematics, weak convergence may refer to: Weak convergence of random variables of a probability distribution; Weak convergence of measures, of a sequence of probability measures; Weak convergence (Hilbert space) of a sequence in a Hilbert space more generally, convergence in weak topology in a Banach space or a topological vector space
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 ...
Weak convergence (Hilbert space) Weak trace-class operator; Wigner's theorem; Y. Nicholas Young (mathematician) This page was last edited on 23 April 2023, at 07:18 ...
The diagram on the right is a summary of the relations, with the arrows pointing from strong to weak. If H is a Hilbert space, the linear space of Hilbert space operators B(X) has a (unique) predual (), consisting of the trace class operators, whose dual is B(X).
Tightness is often a necessary criterion for proving the weak convergence of a sequence of probability measures, especially when the measure space has infinite dimension. See Finite-dimensional distribution; Prokhorov's theorem; Lévy–Prokhorov metric; Weak convergence of measures; Tightness in classical Wiener space; Tightness in Skorokhod space
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.
The ultraweak topology can be obtained from the weak operator topology as follows. If H 1 is a separable infinite dimensional Hilbert space then B(H) can be embedded in B(H⊗H 1) by tensoring with the identity map on H 1. Then the restriction of the weak operator topology on B(H⊗H 1) is the ultraweak topology of B(H).