Search results
Results From The WOW.Com Content Network
If a sequence converges strongly (that is, if it converges in norm), then it converges weakly as well. Since every closed and bounded set is weakly relatively compact (its closure in the weak topology is compact), every bounded sequence in a Hilbert space H contains a weakly convergent subsequence.
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
Starting in the early 1900s, David Hilbert and Marcel Riesz made extensive use of weak convergence. The early pioneers of functional analysis did not elevate norm convergence above weak convergence and oftentimes viewed weak convergence as preferable. [1]
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).
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 ...
A Hilbert space is a vector space equipped with an inner product operation, which allows lengths and angles to be defined. Furthermore, Hilbert spaces are complete, which means that there are enough limits in the space to allow the techniques of calculus to be used. A Hilbert space is a special case of a Banach space.
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).