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6 results

  • 27 - Weak and Weak-* Convergence

  • from Part IV - Banach Spaces
  • James C. Robinson, University of Warwick
  • Book:
    An Introduction to Functional Analysis
    Published online:
    21 March 2020
    Print publication:
    12 March 2020, pp 282-298

  • Summary

    We define weak convergence and give some examples, including a proof of Schur’s Theorem that weak and strong convergence coincide in l^1. We also show that closed convex subsets of Banach spaces are weakly closed. We then introduce weak-* convergence and prove two powerful weak compactness theorems: Helly’s Theorem for weak-* convergence in the duals of separable Banach spaces and a weak sequential compactness theorem in reflexive Banach spaces.

  • Appendix A - Zorn’s Lemma

  • from APPENDICES
  • James C. Robinson, University of Warwick
  • Book:
    An Introduction to Functional Analysis
    Published online:
    21 March 2020
    Print publication:
    12 March 2020, pp 301-304

  • Summary

    We define weak convergence and give some examples, including a proof of Schur’s Theorem that weak and strong convergence coincide in l^1. We also show that closed convex subsets of Banach spaces are weakly closed. We then introduce weak-* convergence and prove two powerful weak compactness theorems: Helly’s Theorem for weak-* convergence in the duals of separable Banach spaces and a weak sequential compactness theorem in reflexive Banach spaces.

  • Appendix B - Lebesgue Integration

  • from APPENDICES
  • James C. Robinson, University of Warwick
  • Book:
    An Introduction to Functional Analysis
    Published online:
    21 March 2020
    Print publication:
    12 March 2020, pp 305-318

  • Summary

    We define weak convergence and give some examples, including a proof of Schur’s Theorem that weak and strong convergence coincide in l^1. We also show that closed convex subsets of Banach spaces are weakly closed. We then introduce weak-* convergence and prove two powerful weak compactness theorems: Helly’s Theorem for weak-* convergence in the duals of separable Banach spaces and a weak sequential compactness theorem in reflexive Banach spaces.

  • Appendix C - The Banach–Alaoglu Theorem

  • from APPENDICES
  • James C. Robinson, University of Warwick
  • Book:
    An Introduction to Functional Analysis
    Published online:
    21 March 2020
    Print publication:
    12 March 2020, pp 319-330

  • Summary

    We define weak convergence and give some examples, including a proof of Schur’s Theorem that weak and strong convergence coincide in l^1. We also show that closed convex subsets of Banach spaces are weakly closed. We then introduce weak-* convergence and prove two powerful weak compactness theorems: Helly’s Theorem for weak-* convergence in the duals of separable Banach spaces and a weak sequential compactness theorem in reflexive Banach spaces.

  • ON INTEGRAL ESTIMATES OF NONNEGATIVE POSITIVE DEFINITE FUNCTIONS

  • Part of
  • ANDREY EFIMOV, MARCELL GAÁL, SZILÁRD GY. RÉVÉSZ
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 96 / Issue 1 / August 2017
    Published online by Cambridge University Press:
    13 March 2017, pp. 117-125
    Print publication:
    August 2017
    • Article
      • You have access
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  • Let $\ell >0$ be arbitrary. We introduce the extremal quantities

    $$\begin{eqnarray}G(\ell ):=\sup _{f}\int _{-\ell }^{\ell }f\,dx\,\bigg/\int _{-1}^{1}f\,dx,\quad C(\ell ):=\sup _{f}\sup _{a\in \mathbb{R}}\int _{a-\ell }^{a+\ell }f\,dx\,\bigg/\int _{-1}^{1}f\,dx,\end{eqnarray}$$
    where the supremum is taken over all not identically zero nonnegative positive definite functions. We investigate how large these extremal quantities can be. This problem was originally posed by Yu. Shteinikov and S. Konyagin (for the case $\ell =2$) and is an extension of the classical problem of Wiener. In this note we obtain exact values for the right limits $\overline{\lim }_{\unicode[STIX]{x1D700}\rightarrow 0+}G(k+\unicode[STIX]{x1D700})$ and $\overline{\lim }_{\unicode[STIX]{x1D700}\rightarrow 0+}C(k+\unicode[STIX]{x1D700})$$(k\in \mathbb{N})$ taken over doubly positive functions, and sufficiently close bounds for other values of $\ell$.

  • AN APPROACH TO CAPABLE GROUPS AND SCHUR’S THEOREM

  • Part of
  • MITRA HASSANZADEH, RASOUL HATAMIAN
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 92 / Issue 1 / August 2015
    Published online by Cambridge University Press:
    05 May 2015, pp. 52-56
    Print publication:
    August 2015
    • Article
      • You have access
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  • Podoski and Szegedy [‘On finite groups whose derived subgroup has bounded rank’, Israel J. Math.178 (2010), 51–60] proved that for a finite group $G$ with rank $r$, the inequality $[G:Z_{2}(G)]\leq |G^{\prime }|^{2r}$ holds. In this paper we omit the finiteness condition on $G$ and show that groups with finite derived subgroup satisfy the same inequality. We also construct an $n$-capable group which is not $(n+1)$-capable for every $n\in \mathbf{N}$.