Baku, Azerbaijan| 41 INTERNATIONAL JOURNAL of ACADEMIC RESEARCH Vol. 5. No. 1. January, 2013 M.A.M. Ferreira, M. Andrade, J.A. Filipe. A note on weak convergence in Hilbert spaces. International Journal of Academic Research Part A; 2013; 5(1), 41-43. DOI: 10.7813/2075-4124.2012/A.7 A NOTE ON WEAK CONVERGENCE IN HILBERT SPACES 1 Prof. Dr. Manuel Alberto M. Ferreira, Prof. Dr. Marina Andrade, Prof. Dr. José António Filipe Instituto Universitário de Lisboa (ISCTE – IUL), BRU - IUL, Lisboa (PORTUGAL) manuel.ferreira@iscte.pt, marina.andrade@iscte.pt, jose.filipe@iscte.pt DOI: 10.7813/2075-4124.2012/A.7 ABSTRACT In the Hilbert spaces domain, it is discussed in this work under which conditions weak convergence implies convergence. Key words: Hilbert spaces, weak convergence, convergence 1. IN HILBERT SPACES WEAK CONVERGENCE IMPLIES CONVERGENCE? It is natural after (1) to pose the following question: - In which conditions weak convergence implies convergence? Begin with the following result: Theorem 1.1 2 Suppose that ݔ ௡ converges weakly for x and ݔ‖ ௡ ‖ for ‖ݔ‖. So ݔ ௡ converges for x. Dem: In fact, it is trivial to observe that ݔ‖ ௡ − ‖ݔ ଶ ݔ‖ = ௡ ‖ ଶ +‖ ‖ݔ ଶ −[ ݔ ௡ ݔ,]−[ ݔ ,ݔ ௡ ]→‖ ‖ݔ ଶ +‖ ‖ݔ ଶ − 2[ݔ ,ݔ]= 2‖ ‖ݔ ଶ − 2‖ ‖ݔ ଶ = 0. So ݔ‖ ௡ − ‖ݔ ଶ →0.■ A result, much more useful than the former one in the applications, on weak convergence is the following: Theorem 1.2 (Banach-Sacks) 3 Suppose that ݔ ௡ converges weakly for x. Then it is possible to determine a subsequence ൛ ݔ ௡ ೖ ൟ such that the arithmetical means ଵ ௠ ∑ ݔ ௡ ೖ ௠ ௞ୀଵ converge for x. Dem: Without loss of generality, it may be supposed that x = 0. Choose ݔ ௡ ೖ in the following way: - ݔ ௡ భ ݔ= ଵ , - Due to the weak convergence, it is possible to choose ݔ ௡ మ , such that ห ݔ ௡ భ ݔ, ௡ మ ൧ห < 1, - Having chosen ݔ ௡ భ , …, ݔ ௡ ೖ it is evident that it is admissible to choose ݔ ௡ ೖశభ such that ห ݔ ௡ ೔ ݔ, ௡ ೖశభ ൧ห < ଵ ௞ ,  = 1, 2, … , . 1 This work was financially supported by FCT through the Strategic Project PEst-OE/EGE/UI0315/2011. 2 ‖∙‖ is, as usual, the symbol for the norm of a vector. 3 [∙,∙] is the symbol for inner product.