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2.1 Example. 2.2 Weak convergence of orthonormal sequences. ... weak convergence in a Hilbert space is the convergence of a sequence of points in the weak topology.
This is the “weak convergence of laws without laws being defined” — except asymptotically. [1] In this case the term weak convergence is preferable (see weak convergence of measures), and we say that a sequence of random elements {X n} converges weakly to X (denoted as X n ⇒ X) if
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
In measure theory Prokhorov's theorem relates tightness of measures to relative compactness (and hence weak convergence) in the space of probability measures. It is credited to the Soviet mathematician Yuri Vasilyevich Prokhorov, who considered probability measures on complete separable metric spaces. The term "Prokhorov’s theorem" is also ...
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.
Despite having the same name as weak convergence in the context of functional analysis, weak convergence of measures is actually an example of weak-* convergence. The definitions of weak and weak-* convergences used in functional analysis are as follows: Let be a topological vector space or Banach space.
In other words, the weak and strong topologies share the same convergent sequences. Note however that weak and strong topologies are always distinct in infinite-dimensional space. Examples
In mathematics, the Opial property is an abstract property of Banach spaces that plays an important role in the study of weak convergence of iterates of mappings of Banach spaces, and of the asymptotic behaviour of nonlinear semigroups. The property is named after the Polish mathematician Zdzisław Opial.