A monomial basis for the holomorphic functions on 𝑐₀

Dineen, Seán; Mujica, Jorge

doi:10.1090/S0002-9939-2012-11436-4

A monomial basis for the holomorphic functions on $c_{0}$
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by Seán Dineen and Jorge Mujica

Proc. Amer. Math. Soc. 141 (2013), 1663-1672

DOI: https://doi.org/10.1090/S0002-9939-2012-11436-4

Published electronically: November 2, 2012

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Abstract:

For over thirty years it has been known that the monomials form a basis for the $n$-homogeneous polynomials on certain infinite dimensional Banach spaces. Recently, Defant and Kalton have shown that these are never unconditional. In this article we show that the monomials form a basis for both the holomorphic functions and the holomorphic functions of bounded type on $c_{0}$, both with their natural topologies.

References

Raymundo Alencar, On reflexivity and basis for $P(^mE)$, Proc. Roy. Irish Acad. Sect. A 85 (1985), no. 2, 131–138. MR 845536
Andreas Defant and Nigel Kalton, Unconditionality in spaces of $m$-homogeneous polynomials, Q. J. Math. 56 (2005), no. 1, 53–64. MR 2124579, DOI 10.1093/qmath/hah022
Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR 737004, DOI 10.1007/978-1-4612-5200-9
Verónica Dimant and Seán Dineen, Banach subspaces of spaces of holomorphic functions and related topics, Math. Scand. 83 (1998), no. 1, 142–160. MR 1662092, DOI 10.7146/math.scand.a-13846
Seán Dineen, Complex analysis in locally convex spaces, Notas de Matemática [Mathematical Notes], vol. 83, North-Holland Publishing Co., Amsterdam-New York, 1981. MR 640093
Seán Dineen, Complex analysis on infinite-dimensional spaces, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 1999. MR 1705327, DOI 10.1007/978-1-4471-0869-6
P. Galindo, M. Maestre, and P. Rueda, Biduality in spaces of holomorphic functions, Math. Scand. 86 (2000), no. 1, 5–16. MR 1738512, DOI 10.7146/math.scand.a-14278
Bogdan C. Grecu and Raymond A. Ryan, Polynomials on Banach spaces with unconditional bases, Proc. Amer. Math. Soc. 133 (2005), no. 4, 1083–1091. MR 2117209, DOI 10.1090/S0002-9939-04-07738-X
Hans Jarchow, Locally convex spaces, Mathematische Leitfäden. [Mathematical Textbooks], B. G. Teubner, Stuttgart, 1981. MR 632257, DOI 10.1007/978-3-322-90559-8
Mário C. Matos, On holomorphy in Banach spaces and absolute convergence of Fourier series, Portugal. Math. 45 (1988), no. 4, 429–450. MR 982911
Jorge Mujica, Complex analysis in Banach spaces, North-Holland Mathematics Studies, vol. 120, North-Holland Publishing Co., Amsterdam, 1986. Holomorphic functions and domains of holomorphy in finite and infinite dimensions; Notas de Matemática [Mathematical Notes], 107. MR 842435
Raymond A. Ryan, Holomorphic mappings on $l_1$, Trans. Amer. Math. Soc. 302 (1987), no. 2, 797–811. MR 891648, DOI 10.1090/S0002-9947-1987-0891648-7
R.A. Ryan, Applications of topological tensor products to infinite dimensional holomorphy, Thesis, Trinity College Dublin, 1980.
Milena Venkova, Global Schauder decompositions of locally convex spaces, Math. Scand. 101 (2007), no. 1, 65–82. MR 2353242, DOI 10.7146/math.scand.a-15032