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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.
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
In analysis, a topology is called strong if it has many open sets and weak if it has few open sets, so that the corresponding modes of convergence are, respectively, strong and weak. (In topology proper, these terms can suggest the opposite meaning, so strong and weak are replaced with, respectively, fine and coarse.)
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.
The predual of B(H) is the trace class operators C 1 (H), and it generates the w*-topology on B(H), called the weak-star operator topology or σ-weak topology. The weak-operator and σ-weak topologies agree on norm-bounded sets in B(H). A net {T α} ⊂ B(H) converges to T in WOT if and only Tr(T α F) converges to Tr(TF) for all finite-rank ...
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 .