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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.
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 ...
Warning: in spite of the name "strongest topology", it is weaker than the norm topology.) The σ-weak topology or ultraweak topology or weak-* operator topology or weak-* topology or weak topology or σ(B(H), B(H) *) topology is defined by the family of seminorms |(w, x)| for elements w of B(H) *. It is stronger than the weak operator topology.
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
Weak topology The weak topology on a set, with respect to a collection of functions from that set into topological spaces, is the coarsest topology on the set which makes all the functions continuous. Weaker topology See Coarser topology. Beware, some authors, especially analysts, use the term stronger topology. Weakly countably compact
Let (,) be a separable metric space.Let () denote the collection of all probability measures defined on (with its Borel σ-algebra).. Theorem. A collection () of probability measures is tight if and only if the closure of is sequentially compact in the space () equipped with the topology of weak convergence.
In a topological abelian group, convergence of a series is defined as convergence of the sequence of partial sums. An important concept when considering series is unconditional convergence, which guarantees that the limit of the series is invariant under permutations of the summands.
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 ]