Vector spaces of entire functions of unbounded type

Codes, Jerónimo

doi:10.1090/S0002-9939-2010-10817-1

Vector spaces of entire functions of unbounded type
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by Jerónimo López - Salazar Codes PDF

Proc. Amer. Math. Soc. 139 (2011), 1347-1360 Request permission

Abstract:

Let $E$ be an infinite dimensional complex Banach space. We prove the existence of an infinitely generated algebra, an infinite dimensional closed subspace and a dense subspace of entire functions on $E$ whose non-zero elements are functions of unbounded type. We also show that the $\tau _{\delta }$ topology on the space of all holomorphic functions cannot be obtained as a countable inductive limit of Fréchet spaces.

References

José M. Ansemil, Richard M. Aron, and Socorro Ponte, Representation of spaces of entire functions on Banach spaces, Publ. Res. Inst. Math. Sci. 45 (2009), no. 2, 383–391. MR 2510506, DOI 10.2977/prims/1241553124
J. M. Ansemil, R. Aron, S. Ponte, Behavior of entire functions on balls in a Banach space, to appear in Indag. Math.
Richard M. Aron, Entire functions of unbounded type on a Banach space, Boll. Un. Mat. Ital. (4) 9 (1974), 28–31 (English, with Italian summary). MR 0374911
Richard M. Aron and M. Schottenloher, Compact holomorphic mappings on Banach spaces and the approximation property, J. Functional Analysis 21 (1976), no. 1, 7–30. MR 0402504, DOI 10.1016/0022-1236(76)90026-4
Richard Aron, V. I. Gurariy, and J. B. Seoane, Lineability and spaceability of sets of functions on $\Bbb R$, Proc. Amer. Math. Soc. 133 (2005), no. 3, 795–803. MR 2113929, DOI 10.1090/S0002-9939-04-07533-1
Richard M. Aron and Juan B. Seoane-Sepúlveda, Algebrability of the set of everywhere surjective functions on $\Bbb C$, Bull. Belg. Math. Soc. Simon Stevin 14 (2007), no. 1, 25–31. MR 2327324
R. M. Aron, F. J. García-Pacheco, D. Pérez-García, and J. B. Seoane-Sepúlveda, On dense-lineability of sets of functions on $\Bbb R$, Topology 48 (2009), no. 2-4, 149–156. MR 2596209, DOI 10.1016/j.top.2009.11.013
Carlos Benítez, Yannis Sarantopoulos, and Andrew Tonge, Lower bounds for norms of products of polynomials, Math. Proc. Cambridge Philos. Soc. 124 (1998), no. 3, 395–408. MR 1636556, DOI 10.1017/S030500419800259X
S. J. Dilworth, Maria Girardi, and W. B. Johnson, Geometry of Banach spaces and biorthogonal systems, Studia Math. 140 (2000), no. 3, 243–271. MR 1784153
Seán Dineen, Holomorphic functions on locally convex topological vector spaces. I. Locally convex topologies on ${\cal H}(U)$, Ann. Inst. Fourier (Grenoble) 23 (1973), no. 1, 19–54 (English, with French summary). MR 500153
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
V. I. Gurariĭ, Subspaces and bases in spaces of continuous functions, Dokl. Akad. Nauk SSSR 167 (1966), 971–973 (Russian). MR 0199674
Lorenz Halbeisen and Norbert Hungerbühler, The cardinality of Hamel bases of Banach spaces, East-West J. Math. 2 (2000), no. 2, 153–159. MR 1825451
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