Volume 5, Number 3 / June issue 2018
Manuel Alberto M. Ferreira
The weak convergence in Hilbert spaces concept construction (revisited)

We intend in this article critically review the weak convergence concept in the real Hilbert spaces domain. Indeed the purpose is to review, correct and enlarge the paper by Ferreira (2014) on this subject, where is studied mainly its construction process. The main goal is to enlarge the Bolzano -Weierstrass theorem field of validity. Then it is discussed in which conditions weak convergence implies convergence.
Keywords: Hilbert spaces; Bolzano-Weierstrass theorem; weak convergence

Cite this article:
Manuel Alberto M. Ferreira. The weak convergence in Hilbert spaces concept construction (revisited). Acta Scientiae et Intellectus, 5(3)2019, 5-12.


  1. H.L. Royden, Real Analysis (3rd edition), New York: MacMilan Publishing Co. Inc., 1968.
  2. J.Y. Campbell, A.W. Lo and A.C. MacKinlay, The Econometrics of Financial Markets, Princeton, New Jersey: Princeton University Press, 1977.
  3. L.V. Kantorovich and G.P. Akilov, Functional Analysis (2nd edition), Oxford: Pergamon Press, 1982.
  4. M.A.M. Ferreira, The Hahn-Banach theorem for the normed spaces in: J. Jafarov, A. Khankishiyev et al (Eds), Emerging Issues in the Natural and Applied Sciences, Azerbaijan: Progress IPS LLC and IJAR, 3(1) (2013), 5-17.
  5. M.A.M. Ferreira, The weak convergence in Hilbert spaces concept construction, Information Sciences and Computing, Volume 2014, Number 2, Article ID ISC260314, 06 pages. Available online at
  6. M.A.M. Ferreira, The minimax theorem as Hahn-Banach theorem consequence. Acta Scientiae et Intellectus, 1(1) (2015), 58-66.
  7. M.A.M. Ferreira and M. Andrade, Riesz representation theorem in Hilbert spaces separation theorems, International Journal of Academic Research, 3(6) (II Part) (2011), 302-304.
  8. M.A.M. Ferreira, M. Andrade and J.A. Filipe, Weak convergence in Hilbert spaces, International Journal of Academic Research, 4(4) (Part A) (2012), 34-36.
  9. R.R. Nigmatullin, J.T. Machado and R. Menezes, Self-similarity principle: The reduced description of randomness, Central European Journal of Physics, 11(6) (2013), 724-739.
  10. S.R. Bentes and R. Menezes, On the predictability of realized volatility using feasible GLS, Journal of Asian Economics, 28(2013), 58-66.