Skip to main content
×
Home

Estimates by polynomials

  • R.M. Aron (a1), Y.S. Choi (a2) and J.G. Llavona (a3)
Abstract

Consider the following possible properties which a Banach space X may have: (P): If (xi) and (yj) are bounded sequences in X such that for all n ≥ 1 and for every continuous n-homogeneous polynomial P on X, P(xj) − (yj) → 0, then Q(xjyj) → 0 for all m ≥ 1 and for every continuous m-homogeneous polynomial Q on X.

(RP): If (xj)and (yj) are bounded sequences in X such that for all n ≥ 1 and for every continuous n-homogeneous polynomial P on X, P(xjyj) → 0, then Q(xj) − Q(yj) → 0 for all m ≥ 1 and for every continuous m-homogeneous polynimial Q on X. We study properties (P) and (RP) and their relation with the Schur proqerty, Dunford-Pettis property, Λ, and others. Several applications of these properties are given.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Estimates by polynomials
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about sending content to Dropbox.

      Estimates by polynomials
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about sending content to Google Drive.

      Estimates by polynomials
      Available formats
      ×
Copyright
References
Hide All
[1]Alencar R., Aron R.M. and Dineen S., ‘A reflexive space of holomorphic functions in infinitely many variables’, Proc. Amer. Math. Soc. 90 (1984), 407411.
[2]Aron R.M. and Galindo P., ‘Weakly compact multilinear mappings’, (preprint).
[3]Aron R.M. and Prolla J.B., ‘Polynomial approximation of differentiable functions on Banach spaces’, J. Reine Agnew. Math. 313 (1980), 195216.
[4]Carne T., Cole B. and Gamelin T., ‘A uniform algebra of analytic functions on a Banach space’, Trans. Amer. Math. Soc. 314 (1989), 639659.
[5]Castillo J.F. and Sanchez C., ‘Weakly-p-compact, p-Banach-Saks and super-reflexive Banach spaces’, J. Math. Anal. Appl. 185 (1994), 256261.
[6]Choi Y.S. and Kim S.G., ‘Polynomial properties of Banach spaces’, J. Math. Anal. Appl. 190 (1995), 203210.
[7]Davie A.M. and Gamelin T.W., ‘A theorem on polynomial-star approximation’, Proc. Amer. Math. Soc. 106 (1989), 351358.
[8]Diestel J., ‘A survey of results related to the Dunford-Pettis property’, in Contemp. Math. 2 (Amer. Math. Soc., Providence, R.I., 1980), pp. 1560.
[9]Diestel J., Geometry of Banach spaces, Lecture Notes in Mathematics 485 (Springer-Verlag, Berlin, Heidelberg, New York, 1975).
[10]Van Dulst D., Reflexive and super-reflexive spaces, Math. Centre Tracts 102 (Amsterdam, 1982).
[11]Dunford N. and Schwartz J.T., Linear Operators, Part I, General Theory (J. Wiley, New York, 1964).
[12]Farmer J.D., ‘Polynomial reflexivity in Banach spaces’, Israel J. Math. 87 (1994), 257273.
[13]Farmer J.D. and Johnson W.B., ‘Polynomial Schur and polynomial Dunford-Pettis properties’, in Proc. Intern. Research Workshop on Banach Space Theory (Mérida, Venezuela), (Johnson W.B. and Lin B.L., Editors) (Amer. Math. Soc., Providence, RI, 1993), pp. 95105.
[14]Jaramillo J.A. and Prieto A., ‘Weak-polynomial convergence on a Banach space’, Proc. Amer. Math. Soc. 118 (1993), 463468.
[15]Josefson B., ‘Bounding subsets of ℓ (A)’, J. Math. Pures Appl. 57 (1978), 397421.
[16]Grothendieck A., ‘Sur les applications linéaires faiblement compactes d'espaces du type C(K)’, Canad. J. Math. 5 (1953), 129173.
[17]Pelczynski A., ‘A property of multilinear operations’, Studia Math. 16 (1957), 173182.
[18]Petunin Y.I. and Savkin V.I., ‘Convergence generated by analytic functions’, Ukranian. Math. J. 40 (1988), 676679.
[19]Rosenthal H.P., ‘Some recent discoveries in the isomorphic theory of Banach spaces’, Bull. Amer. Math. Soc. 84 (1980), 803831.
[20]Ryan R.A., ‘Dunford-Pettis properties’, Bull. Acad. Polon. Sci. Math. 27 (1979), 373379.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 15 *
Loading metrics...

Abstract views

Total abstract views: 34 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 19th November 2017. This data will be updated every 24 hours.