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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 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
This definition of weak convergence can be extended for any metrizable topological space. It also defines a weak topology on (), the set of all probability measures defined on (,). The weak topology is generated by the following basis of open sets:
(Warning: the weak Banach space topology and the weak operator topology and the ultraweak topology are all sometimes called the weak topology, but they are different.) The Mackey topology or Arens-Mackey topology is the strongest locally convex topology on B( H ) such that the dual is B( H ) * , and is also the uniform convergence topology on ...
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
The corresponding topology is called weak topology and is the initial topology of all bounded continuous functions on . The weak topology corresponds to the weak* topology in functional analysis. If X {\displaystyle X} is also separable , the weak convergence is metricized by the Lévy–Prokhorov metric .
Convergence spaces generalize the notions of convergence that are found in point-set topology, including metric convergence and uniform convergence. Every topological space gives rise to a canonical convergence but there are convergences, known as non-topological convergences , that do not arise from any topological space. [ 1 ]