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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.
Definition. The weak topology on X induced by Y (and b) is the weakest topology on X, denoted by 𝜎(X, Y, b) or simply 𝜎(X, Y), making all maps b(•, y) : X → continuous, as y ranges over Y. [1] The weak topology on Y is now automatically defined as described in the article Dual system. However, for clarity, we now repeat it.
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:
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
(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 ...
The definition of convergence in distribution may be extended from random vectors to more general random elements in arbitrary metric spaces, and even to the “random variables” which are not measurable — a situation which occurs for example in the study of empirical processes. This is the “weak convergence of laws without laws being ...
This concept is often contrasted with uniform convergence.To say that = means that {| () |:} =, where is the common domain of and , and stands for the supremum.That is a stronger statement than the assertion of pointwise convergence: every uniformly convergent sequence is pointwise convergent, to the same limiting function, but some pointwise convergent sequences are not uniformly convergent.
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