# Holomorphic functions and Banach-nuclear decompositions of Fréchet spaces

Studia Mathematica (1995)

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

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topDineen, 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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- [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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