Weak convergence in Hilbert spaces

2012

In the Hilbert spaces domain, it is discussed in this work under which conditions weak convergence implies convergence.

International Journal of Pure and Applied Mathematics

We establish weak (strong) convergence of Ishikawa iterates of two asymptotically (quasi-)nonexpansive maps without any condition on the rate of convergence associated with the two maps. Moreover, our weak convergence results do not require any of the Opial condition, Kadec-Klee property or Fréchet differentiable norm.

This paper provides a sufficient condition to guarantee the stability of weak limits under nonlinear operators acting on vector-valued Lebesgue spaces. This nonlinear framework places the weak convergence in perspective. Such an approach allows short and insightful proofs of important results in Functional Analysis such as: weak convergence in L ∞ implies strong convergence in L p for all 1 ≤ p < ∞, weak convergence in L 1 v.s. strong convergence in L 1 and the Brezis-Lieb theorem. The final goal is to use this framework as a strategy to grapple with a nonlinear weak spectral problem on W 1,p .

Proceedings of the American Mathematical Society, 1978

Let A" be any topological space, and C(X) the space of bounded continuous functions on X. We give a nonstandard characterization of weak convergence of a net of bounded linear functionals on CiX) to a tight Baire measure on X. This characterization applies whether or not the net or the individual functionals in the net are tight. Moreover, the characterization is expressed in terms of the values of an associated net of countably additive measures on all Baire sets of X; no distinguished family, such as the family of continuity sets of the limit, is involved. As a corollary, we obtain a new proof that a tight set of measures is relatively weakly compact.

We discuss some basic properties of polar convergence in metric spaces. Polar convergence is closely connected with the notion of Delta-convergence of T.C. Lim known for several years. Possible existence of a topology which induces polar convergence is also investigated. Some applications of polar convergence follow.

We discuss the notions of strong convergence and weak convergence in n-inner product spaces and study the relation between them. In particular, we show that the strong convergence implies the weak convergence and disprove the converse through a counterexample , by invoking an analogue of Parseval's identity in n-inner product spaces.

Applied Mathematics Letters, 2011

We consider an implicit iterative process for two finite families of mappings in a real Banach space and prove strong convergence results without using the Lipschitz condition on mappings. Our results mainly improve and extend the recent results of Chang et al.

Anais Da Academia Brasileira De Ciencias, 2003

In this paper, we prove that if a Nemytskii operator maps Lp( , E) into Lq( , F), for p, q greater than 1, E, F separable Banach spaces and F reflexive, then a sequence that converge weakly and a.e. is sent to a weakly convergent sequence. We give a counterexample proving that if q = 1 and p is greater than 1 we may not have weak sequential continuity of such operator. However, we prove that if p = q = 1, then a weakly convergent sequence that converges a.e. is mapped into a weakly convergent sequence by a Nemytskii operator. We show an application of the weak continuity of the Nemytskii operators by solving a nonlinear functional equation on W1,p( ), providing the weak continuity of some kind of resolvent operator associated to it and getting a regularity result for such solution.

Stochastic Processes and their Applications, 1997

We discuss three forms of convergence in distribution which are stronger than the normal weak convergence. They have the advantage that they are non-topological in nature and are inherited by discontinuous functions of the original random variables-clearly an improvement on 'normal' weak convergence. We give necessary and sufficient conditions for the three types of convergence and go on to give some applications which are very hard to prove in a more restricted setting.