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Polynomials and limited sets


Author: Pablo Galindo
Journal: Proc. Amer. Math. Soc. 124 (1996), 1481-1488
MSC (1991): Primary 46G20; Secondary 46B20
DOI: https://doi.org/10.1090/S0002-9939-96-03175-9
MathSciNet review: 1307518
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Abstract: We prove that scalar-valued polynomials are weakly continuous on limited sets and that, as in the case of linear mappings, every $c_0$-valued polynomial maps limited sets into relatively compact ones. We also show that a scalar-valued polynomial whose derivative is limited is weakly sequentially continuous.


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  • 1. R. Aron, B. Cole, and T. Gamelin, Weak-star continuous analytic functions, Canad. J. Math. 47 (1995), 673--683.
  • 2. R. Aron, C. Hervés, and M. Valdivia, Weakly continuous mappings on Banach spaces, J. Funct. Anal. 52, (1983), 189--204. MR 84g:46066
  • 3. J. Bourgain and J. Diestel, Limited operators and strict cosingularity, Math. Nachr. 11, (1984), 55--58. MR 86d:47024
  • 4. J. Dieudonné, Sur les espaces de Köthe, J. Analyse Math. 1, (1951), 81--115. MR 12:834b
  • 5. S. Dineen, Complex analysis in locally convex spaces, North-Holland Math. Stud., vol. 57, North-Holland, Amsterdam and New York, 1980. MR 84b:46050
  • 6. J. Diestel, Sequences and series in Banach spaces, Graduate Texts in Math., vol. 92, Springer-Verlag, Berlin and New York, 1984. MR 85i:46020
  • 7. L. Drewnowski, On Banach spaces with the Gelfand-Phillips property I, Math. Z. 19, (1986), 405--411. MR 88d:46024
  • 8. L. Drenowski and G. Emmanuele, On Banach spaces with the Gelfand-Phillips property II, Rend. Cir. Mat. Palermo Ser. II 38, (1989), 377--391. MR 92a:46016
  • 9. J. Farmer and W. Johnson, Polynomial Schur and polynomial Dunford-Pettis properties, Contemp. Math., vol. 144, Amer. Math. Soc., Providence, RI, 1993, pp. 95--105. MR 94e:46078
  • 10. M. González and J. Gutiérrez, Unconditionally converging polynomials on Banach spaces, Math. Proc. Cambridge Philos. Soc. 117 (1995), 321--331. CMP95:05
  • 11. ------, The compact weak topology on a Banach space, Proc. Roy. Soc. Edinburgh Sect. A 12, (1992), 367--379. MR 93c:46022
  • 12. B. Josefson, Bounding sets in $l^\infty(A)$, J. Math. Pures Appl. 57, (1975), 397--421. MR 81a:46019
  • 13. ------, A Banach space containing non-trivial limited sets but no non-trivial bounding set, Israel J. Math. 71, (1990), 321--327. MR 92g:46018
  • 14. D. Leung, A Gelfand-Phillips property with respect to the weak topology, Math. Nachr. 14, (1990), 177--181. MR92j:46032
  • 15. M. Lindström, On compact and bounding holomorphic mappings, Proc. Amer. Math. Soc. 10, (1989), 356--361. MR 89f:46103
  • 16. ------, A characterization of Schwartz spaces, Math. Z. 19, (1988), 423--430. MR 89f:46009
  • 17. M. Lindström and T. Schlumprecht, On limitedness on locally convex spaces, Arch. Math. 53, (1988), 65--74. MR 90f:46003
  • 18. J. Mujica, Linearization of bounded holomorphic mappings on Banach spaces, Trans. Amer. Math. Soc. 32, (1991), 867--887. MR 91h:46088
  • 19. T. Schlumprecht, A limited set that is not bounding, Proc. Roy. Irish Acad. Sect. A 90, (1990), 125--129. MR 93b:46032

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Additional Information

Pablo Galindo
Email: galindo@vm.ci.uv.es

DOI: https://doi.org/10.1090/S0002-9939-96-03175-9
Keywords: Polynomial, limited set, weakly conditionally compact set
Received by editor(s): June 13, 1994
Received by editor(s) in revised form: October 19, 1994
Additional Notes: The author was supported in part by DGICYT pr. 91-0326 and by grant 93-081
Communicated by: Theodore Gamelin
Article copyright: © Copyright 1996 American Mathematical Society

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