Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Vector spaces of entire functions of unbounded type


Author: Jerónimo López - Salazar Codes
Journal: Proc. Amer. Math. Soc. 139 (2011), 1347-1360
MSC (2010): Primary 46G20; Secondary 46E50
DOI: https://doi.org/10.1090/S0002-9939-2010-10817-1
Published electronically: November 23, 2010
MathSciNet review: 2748427
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. J. M. Ansemil, R. M. Aron, S. Ponte, Representation of spaces of entire functions on Banach spaces, Publ. Res. Inst. Math. Sci., 45 (2009), 383-391. MR 2510506 (2010c:46105)
  • 2. J. M. Ansemil, R. Aron, S. Ponte, Behavior of entire functions on balls in a Banach space, to appear in Indag. Math.
  • 3. R. M. Aron, Entire functions of unbounded type on a Banach space, Boll. Unione Mat. Ital., 9 (1974), 28-31. MR 0374911 (51:11107)
  • 4. R. M. Aron, M. Schottenloher, Compact holomorphic mappings on Banach spaces and the approximation property, J. Funct. Anal., 21 (1976), 7-30. MR 0402504 (53:6323)
  • 5. R. M. Aron, V. I. Gurariy, J. B. Seoane Sepúlveda, Lineability and spaceability of sets of functions on $ \mathbb{R}$, Proc. Amer. Math. Soc., 133 (2005), 795-803. MR 2113929 (2006i:26004)
  • 6. R. M. Aron, J. B. Seoane Sepúlveda, Algebrability of the set of everywhere surjective functions on $ \mathbb{C}$, Bull. Belg. Math. Soc. Simon Stevin, 14 (2007), 25-31. MR 2327324 (2008d:26016)
  • 7. R. M. Aron, F. J. García Pacheco, D. Pérez García, J. B. Seoane Sepúlveda, On dense-lineability of sets of functions on $ \mathbb{R}$, Topology, 48 (2009), 149-156. MR 2596209
  • 8. C. Benítez, Y. Sarantopoulos, A. Tonge, Lower bounds for norms of products of polynomials, Math. Proc. Cambridge Philos. Soc., 124 (1998), pp. 395-408. MR 1636556 (99h:46077)
  • 9. S. J. Dilworth, M. Girardi, W. B. Johnson, Geometry of Banach spaces and biorthogonal systems, Studia Math., 140 (2000), 243-271. MR 1784153 (2001i:46013)
  • 10. S. Dineen, Holomorphic functions on locally convex topological vector spaces. I, Locally convex topologies on $ \mathcal{H}\left( U\right)$, Ann. Inst. Fourier (Grenoble), 23 (1973), 19-54. MR 0500153 (58:17843)
  • 11. S. Dineen, Complex Analysis on Infinite Dimensional Spaces, Springer Monogr. Math., 1999. MR 1705327 (2001a:46043)
  • 12. V. I. Gurariĭ, Subspaces and bases in spaces of continuous functions, Dokl. Akad. Nauk SSSR, 167 (1966), 971-973. MR 0199674 (33:7817)
  • 13. L. Halbeisen, N. Hungerbühler, The cardinality of Hamel bases of Banach spaces, East-West J. Math., 2 (2000), 153-159. MR 1825451 (2002f:46013)
  • 14. J. Mujica, Complex Analysis in Banach Spaces, North-Holland Math. Stud., 120, Amsterdam, 1986. MR 842435 (88d:46084)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 46G20, 46E50

Retrieve articles in all journals with MSC (2010): 46G20, 46E50


Additional Information

Jerónimo López - Salazar Codes
Affiliation: Departamento de Análisis Matemático, Universidad Complutense de Madrid, 28040 Madrid, Spain
Email: jlopezsalazar@mat.ucm.es

DOI: https://doi.org/10.1090/S0002-9939-2010-10817-1
Keywords: Entire function, \it lineable set, $\tau_{\delta}$ topology
Received by editor(s): February 22, 2010
Received by editor(s) in revised form: April 7, 2010
Published electronically: November 23, 2010
Additional Notes: The author was supported by UCM Grant #BE46/08.
Communicated by: Mei-Chi Shaw
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society