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Results in Mathematics

, 74:12 | Cite as

A Letter Concerning Leonetti’s Paper ‘Continuous Projections Onto Ideal Convergent Sequences’

  • Tomasz KaniaEmail author
Article
  • 38 Downloads

Abstract

Leonetti proved that whenever \({\mathcal {I}}\) is an ideal on \({\mathbb {N}}\) such that there exists an uncountable family of sets that are not in \({\mathcal {I}}\) with the property that the intersection of any two distinct members of that family is in \({\mathcal {I}}\), then the space \(c_{0,{\mathcal {I}}}\) of sequences in \(\ell _\infty \) that converge to 0 along \({\mathcal {I}}\) is not complemented. We provide a shorter proof of a more general fact that the quotient space \(\ell _\infty / c_{0,{\mathcal {I}}}\) does not even embed into \(\ell _\infty \).

Keywords

Convergence along an ideal complemented subspace Phillips–Sobczyk theorem Grothendieck space 

Mathematics Subject Classification

Primary 46B20 46B26 Secondary 40A35 

Notes

References

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    Albiac, F., Kalton, N.J.: Topics in Banach Space Theory, Graduate Texts in Mathematics, vol. 233. Springer, New York (2006)zbMATHGoogle Scholar
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    Grothendieck, A.: Sur les applications linéaires faiblement compactes d’espaces du type \(C(K)\). Can. J. Math. 5, 129–173 (1953)CrossRefGoogle Scholar
  3. 3.
    Leonetti, P.: Continuous projections onto ideal convergent sequences. Results Math. 73, 114 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Whitley, R.: Projecting \(m\) onto \(c_0\). Am. Math. Mon. 73, 285–286 (1966)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Institute of MathematicsCzech Academy of SciencesPrague 1Czech Republic

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