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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 ...
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.
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.
In this example, our original Hilbert space is actually the 3-dimensional Euclidean space equipped with the standard scalar product (,) =, our 3-by-3 matrix defines the bilinear form (,) =, and the right-hand-side vector defines the bounded linear functional =. The columns
In the mathematical discipline of functional analysis, the concept of a compact operator on Hilbert space is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators are precisely the closure of finite-rank operators (representable by finite-dimensional matrices) in the topology induced by the operator norm.