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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
Download as PDF; Printable version; In other projects Wikimedia Commons; Wikidata item; ... Weak convergence (Hilbert space) Weak trace-class operator; Wigner's ...
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 ...
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).
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.
where H(D) is the space of holomorphic functions in D. Then L 2,h (D) is a Hilbert space: it is a closed linear subspace of L 2 (D), and therefore complete in its own right. This follows from the fundamental estimate, that for a holomorphic square-integrable function ƒ in D