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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 ...
The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that, see List of general topology topics and Topological property .
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 ...
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
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.
Such a topology is called a vector topology and every topological vector space has a uniform topological structure, allowing a notion of uniform convergence and completeness. Some authors also require that the space is a Hausdorff space (although this article does not).