Search results
Results From The WOW.Com Content Network
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 ...
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 ...
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
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.
If the definition depends on the order of and (e.g. the definition of "the weak topology (,) defined on by ") then by switching the order of and , it is meant that this definition should be applied to (,, ^) (e.g. this gives us the definition of "the weak topology (,) defined on by ").