Holomorphic functions and Banach-nuclear decompositions of Fréchet spaces

Seán Dineen

Studia Mathematica (1995)

  • Volume: 113, Issue: 1, page 43-54
  • ISSN: 0039-3223

Abstract

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We introduce a decomposition of holomorphic functions on Fréchet spaces which reduces to the Taylor series expansion in the case of Banach spaces and to the monomial expansion in the case of Fréchet nuclear spaces with basis. We apply this decomposition to obtain examples of Fréchet spaces E for which the τ_{ω} and τ_{δ} topologies on H(E) coincide. Our result includes, with simplified proofs, the main known results-Banach spaces with an unconditional basis and Fréchet nuclear spaces with DN [2, 4, 5, 6] - together with new examples, e.g. Banach spaces with an unconditional finite-dimensional Schauder decomposition and certain Fréchet-Schwartz spaces. This gives the first examples of Fréchet spaces, which are not nuclear, with τ_{0} = τ_{δ} on H(E).

How to cite

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Dineen, Seán. "Holomorphic functions and Banach-nuclear decompositions of Fréchet spaces." Studia Mathematica 113.1 (1995): 43-54. <http://eudml.org/doc/216158>.

@article{Dineen1995,
abstract = {We introduce a decomposition of holomorphic functions on Fréchet spaces which reduces to the Taylor series expansion in the case of Banach spaces and to the monomial expansion in the case of Fréchet nuclear spaces with basis. We apply this decomposition to obtain examples of Fréchet spaces E for which the τ\_\{ω\} and τ\_\{δ\} topologies on H(E) coincide. Our result includes, with simplified proofs, the main known results-Banach spaces with an unconditional basis and Fréchet nuclear spaces with DN [2, 4, 5, 6] - together with new examples, e.g. Banach spaces with an unconditional finite-dimensional Schauder decomposition and certain Fréchet-Schwartz spaces. This gives the first examples of Fréchet spaces, which are not nuclear, with τ\_\{0\} = τ\_\{δ\} on H(E).},
author = {Dineen, Seán},
journal = {Studia Mathematica},
keywords = {decomposition of holomorphic functions on Fréchet spaces; Taylor series expansion; monomial expansion; Fréchet nuclear spaces with basis; Banach spaces with an unconditional basis},
language = {eng},
number = {1},
pages = {43-54},
title = {Holomorphic functions and Banach-nuclear decompositions of Fréchet spaces},
url = {http://eudml.org/doc/216158},
volume = {113},
year = {1995},
}

TY - JOUR
AU - Dineen, Seán
TI - Holomorphic functions and Banach-nuclear decompositions of Fréchet spaces
JO - Studia Mathematica
PY - 1995
VL - 113
IS - 1
SP - 43
EP - 54
AB - We introduce a decomposition of holomorphic functions on Fréchet spaces which reduces to the Taylor series expansion in the case of Banach spaces and to the monomial expansion in the case of Fréchet nuclear spaces with basis. We apply this decomposition to obtain examples of Fréchet spaces E for which the τ_{ω} and τ_{δ} topologies on H(E) coincide. Our result includes, with simplified proofs, the main known results-Banach spaces with an unconditional basis and Fréchet nuclear spaces with DN [2, 4, 5, 6] - together with new examples, e.g. Banach spaces with an unconditional finite-dimensional Schauder decomposition and certain Fréchet-Schwartz spaces. This gives the first examples of Fréchet spaces, which are not nuclear, with τ_{0} = τ_{δ} on H(E).
LA - eng
KW - decomposition of holomorphic functions on Fréchet spaces; Taylor series expansion; monomial expansion; Fréchet nuclear spaces with basis; Banach spaces with an unconditional basis
UR - http://eudml.org/doc/216158
ER -

References

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  1. [1] J. Bonet and J. C. Díaz, The problem of topologies of Grothendieck and the class of Fréchet T-spaces, Math. Nachr. 150 (1991), 109-118. Zbl0754.46043
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  3. [3] S. Dineen, Bounding subsets of a Banach space, Math. Ann. 192 (1971), 61-70. Zbl0202.12803
  4. [4] S. Dineen, Holomorphic functions on ( c 0 , X b ) -modules, ibid. 196 (1972), 106-116. Zbl0219.46021
  5. [5] S. Dineen, Complex Analysis in Locally Convex Spaces, North-Holland Math. Stud. 57, North-Holland, 1981. Zbl0484.46044
  6. [6] S. Dineen, Analytic functionals on fully nuclear spaces, Studia Math. 73 (1982), 11-32. Zbl0532.46022
  7. [7] S. Dineen, Holomorphic functions and the (BB)-property, Math. Scand., to appear. Zbl0870.46029
  8. [8] N. J. Kalton and N. T. Peck, Twisted sums of sequence spaces and the three space problem, Trans. Amer. Math. Soc. 255 (1979), 1-30. Zbl0424.46004
  9. [9] T. Ketonen, On unconditionality in L p -spaces, Ann. Acad. Sci. Fenn. Dissertationes 35 (1981). 
  10. [10] G. Metafune, Quojections and finite dimensional decompositions, preprint. 
  11. [11] D. Vogt, Subspaces and quotients of (s), in: Functional Analysis: Surveys and Recent Results, K. D. Bierstedt and B. Fuchsteiner (eds.), North-Holland Math. Stud. 27, North-Holland, 1977, 167-187. 

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