Complutense University Library

Polynomial compactness in Banach spaces

Biström, Peter and Jaramillo Aguado, Jesús Ángel and Lindström, Mikael (1998) Polynomial compactness in Banach spaces. Rocky Mountain Journal of Mathematics, 28 (4). pp. 1203-1226. ISSN 0035-7596

[img] PDF
Restricted to Repository staff only until 2020.


Official URL:

View download statistics for this eprint

==>>> Export to other formats


We investigate infinite dimensional Banach spaces equipped with the initial topology with respect to the continuous polynomials. We show nonlinear properties for this topology in both the real and the complex case. A new property for Banach spaces, polynomial Dunford-Pettis property, is introduced. For spaces with this property the compact sets in the topology induced by the polynomials are shown to be invariant under the summation map. For most real Banach spaces we characterize the polynomially compact sets as the bounded sets that are separated from zero by the positive polynomials.

Item Type:Article
Uncontrolled Keywords:Polynomial compactness; Dunford-Pettis property; nonlinear topology
Subjects:Sciences > Mathematics > Functional analysis and Operator theory
ID Code:16735

R. Alencar, R. Aron and S. Dineen, A reflexive space of holomorphic functions in infinite many variables, Proc. Amer. Math. Soc. 90 (1984), 407- 411.

R.M. Aron, Y.S. Choi and J.G. Llavona, Estimates by polynomials, preprint.

R.M. Aron, B.J. Cole and T.W. Gamelin, Spectra of algebras of analytic functions on a Banach space, J. Reine Angew. Math. 415 (1991), 51- 93.

P. Biström, J.A. Jaramillo and M. Lindstr¨om, Algebras of real analytic functions; Homomorphisms and bounding sets, StudiaMath. 115 (1) (1995), 23- 37.

F. Bombal, Operators on vector sequence spaces, Geometric aspects of Banach spaces, London Math. Soc. Lecture Notes Ser. 140 (1989), 94- 106.

J.K. Brooks and P.W. Lewis, Operators on continuous function spaces and convergence in spaces of operators, Adv. Math. 29 (1978), 157 -177.

T.K. Carne, B. Cole and T.W. Gamelin, A uniform algebra of analytic functions on a Banach space, Trans. Amer. Math. Soc. 314 (1989), 639 -659.

J.F. Castillo, On weakly-p-summable sequences in p(X) and C(K,X), preprint.

J.F. Castillo and F. Sánchez, Dunford-Pettis-like properties of continuous vector function spaces, Rev. Mat. Univ. Complut. Madrid 6 (1993), 43- 59.

---- Weakly-p-compact, p-Banach-Saks and super-reflexive Banach spaces, J. Math. Anal. Appl. 185 (1994), 256- 261.

R. Deville, G. Godefroy and V. Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs Surveys Pure Appl. Math. 64 -1993.

J. Diestel, Sequences and series in Banach spaces, Graduate Texts in Math. 92 -1984.

S. Dineen, Holomorphy types on Banach spaces, Studia Math. 39 (1971), 240- 288.

M. Fabian, D. Preiss, H.M. Whitfield and V.E. Zizler, Separating polynomials on Banach spaces, Quart. J. Math. Oxford 40 (1989), 409 -422.

J. Farmer, Polynomial reflexivity in Banach spaces, Israel J.Math. 87 (1994), 257 273.

J. Farmer and W.B. Johnson, Polynomial Schur and polynomial Dunford- Pettis properties, Contem. Math. 144 (1993), 95- 105.

K. Floret, Weakly compact set, Lecture Notes in Math. 801 -1980.

M. Garrido, J. G´omzez and J.A. Jaramillo, Homomorphisms on function algebras, Canad. J. Math. 46 (1994), 734 -745.

M. González and J.M. Gutiérrez, Gantmacher type theorems for holomorphic mappings, preprint.

M. Gonzalo and J.A. Jaramillo, Compact polynomials between Banach spaces, preprint.

R. Gonzalo, Suavidad y polinomios en espacios de Banach, Ph.D. Thesis, Universidad Complutense de Madrid, 1994.

P. Harmand, D. Werner and W. Werner, M-ideals in Banach spaces and Banach algebras, Lecture Notes in Math. 1547, 1993.

R.C. James, Super-reflexive spaces with bases, Pacific J. Math. 41 (1972), 409 -419.

J.A. Jaramillo and A. Prieto, The weak-polynomial convergence on a Banach space, Proc. Amer. Math. Soc. 118 (1993), 463 -468.

B. Josefson, Bounding subsets of l∞(A), J. Math. Pures Appl. (9) 57 (1978), 397-421.

H. Knaust and E. Odell, Weakly null sequences with upper p-estimates, Lecture Notes in Math. 1047 (1991), 85 -107.

L.G. Llavona, Approximation of continuously differentiable functions, North- Holland, Amsterdam, 1986.

J. Mujica, Complex homomorphisms of the algebras of holomorphic functions on Fréchet spaces, Math. Ann. 241 (1979), 73- 82.

G. Pisiser, Factorization of linear operators and geometry of Banach spaces, Amer. Math. Soc. 60, 1986.

H.P. Rosenthal, On quasi-complemented subspaces of Banach spaces, with an appendix on compactness of operators from Lp(μ) to Lp(ν), J. Funct. Anal. 4 (1969), 176- 214.

Deposited On:18 Oct 2012 10:38
Last Modified:07 Feb 2014 09:34

Repository Staff Only: item control page