Arkiv för Matematik

  • Ark. Mat.
  • Volume 38, Number 2 (2000), 343-354.

To what extent does the dual Banach space E′ determine the polynomials over E?

Silvia Lassalle and Ignacio Zalduendo

Full-text: Open access


We show that under conditions of regularity, if E′ is isomorphic to F′, then the spaces of homogeneous polynomials over E and F are isomorphic. Some subspaces of polynomials more closely related to the structure of dual spaces (weakly continuous, integral, extendible) are shown to be isomorphic in full generality.

Article information

Ark. Mat., Volume 38, Number 2 (2000), 343-354.

Received: 10 August 1998
First available in Project Euclid: 31 January 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

2000 © Institut Mittag-Leffler


Lassalle, Silvia; Zalduendo, Ignacio. To what extent does the dual Banach space E′ determine the polynomials over E ?. Ark. Mat. 38 (2000), no. 2, 343--354. doi:10.1007/BF02384324.

Export citation


  • [AB] Aron, R. and Berner, P., A Hahn-Banach extension theorem for analytic mappings, Bull. Soc. Math. France 106 (1978), 3–24.
  • [ACG] Aron, R., Cole, B. and Gamelin, T., Spectra of algebras of analytic functions on a Banach space, J. Reine Angew. Math. 415 (1991), 51–93.
  • [AG] Aron, R. and Galindo, P., Weakly compact multilinear mappings, Proc. Edinburgh Math. Soc. 40 (1997), 181–192.
  • [AGGM] Aron, R., Galindo, P., García, D. and Maestre, M., Regularity and algebras of analytic functions in infinite dimensions, Trans. Amer. Math. Soc. 348 (1996), 543–559.
  • [AHV] Aron, R., Hervés, C. and Valdivia, M., Weakly continuous mappings on Banach spaces, J. Funct. Anal. 52 (1983), 189–204.
  • [CCG] Cabello Sánchez, F., Castillo, J. M. F. and García, R., Polynomials on dual-isomorphic spaces, Ark. Mat. 38 (2000) 37–44.
  • [CDDL] Carando, D., Dimant, V., Duarte, B. and Lassalle, S., K-bounded polynomials, Math. Proc. R. Ir. Acad. 98A (1998), 159–171.
  • [CZ] Carando, D. and Zalduendo, I., A Hahn-Banach theorem for integral polynomials, Proc. Amer. Math. Soc. 127 (1999), 241–250.
  • [DD] Díaz, J. C. and Dineen, S., Polynomials on stable spaces, Ark. Mat. 36 (1998), 87–96.
  • [D] Dineen, S., Complex Analysis in Locally Convex Spaces, Math. Studies 57, North-Holland, Amsterdam, 1981.
  • [FGL] Ferrera, J., Gómez, J. and Llavona, J., On completion of spaces of weakly continuous functions, Bull. London Math. Soc. 15 (1983), 260–264.
  • [GI] Godefroy, G. and Iochum, B., Arens-regularity on Banach algebras and geometry of Banach spaces, J. Funct. Anal. 80 (1988), 47–59.
  • [GJL] Gutiérrez, J., Jaramillo, J. and Llavona, J., Polynomials and geometry of Banach spaces, Extracta Math. 10 (1995), 79–114.
  • [KR] Kirwan, P. and Ryan, R., Extendibility of homogeneous polynomials on Banach spaces, Proc. Amer. Math. Soc. 126 (1998), 1023–1029.
  • [M] Mujica, J., Complex Analysis in Banach Spaces, Math. Studies 120, North-Holland, Amsterdam, 1986.
  • [Ry] Ryan, R., Dunford-Pettis properties, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), 373–379.
  • [S] Stegall, C., Duals of certain spaces with the Dunford-Pettis property, Notices Amer. Math. Soc. 19 (1972), A799.
  • [T] Toma, E., Aplicaçoes holomorfas e polinomios τ-continuous, Thesis, Universidade Federal do Rio de Janeiro, Rio de Janeiro, 1993.
  • [Ü] Ülger, A., Weakly compact bilinear forms and Arens-regularity, Proc. Amer. Math. Soc. 101 (1987), 697–704.
  • [Z1] Zalduendo, I., A canonical extension for analytic functions on Banach spaces, Trans. Amer. Math. Soc. 320 (1990), 747–763.
  • [Z2] Zalduendo, I., Extending polynomials—a survey, Publ. Dep. Análisis Mat., Univ. Complut. Madrid, Secc. 1, No. 41, 1998.