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The Exponential Representation of Holomorphic Functions of Uniformly Bounded Type

Part of: Topological linear spaces and related structures Holomorphic functions of several complex variables Linear function spaces and their duals

Published online by Cambridge University Press:  09 April 2009

Thai Thuan Quang
Thai Thuan Quang
Affiliation:
Department of Mathematics, Pedagogical Institute of Quynhon170 An Duong Vuong, Quynhon, Binhdinh, Vietnam e-mail: tthuanquang@hotmail.com
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Abstract

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It is shown that if E, F are Fréchet spaces, E ∈ (Hub), F ∈ (DN) then H(E, F) = Hub(E, F) holds. Using this result we prove that a Fréchet space E is nuclear and has the property (Hub) if and only if every entire function on E with values in a Fréchet space F ∈ (DN) can be represented in the exponential form. Moreover, it is also shown that if H(F*) has a LAERS and E ∈ (Hub) then H(E × F*) has a LAERS, where E, F are nuclear Fréchet spaces, F* has an absolute basis, and conversely, if H(E × F*) has a LAERS and F ∈ (DN) then E ∈ (Hub).

Keywords

holomorphic functionsholomorphic functions of uniform bounded typetopological linear invariantsexponential representation

MSC classification

Secondary: 32A10: Holomorphic functions 32B10: Germs of analytic sets, local parametrization 46A63: Topological invariants ((DN), ($Omega$), etc.) 46E50: Spaces of differentiable or holomorphic functions on infinite-dimensional spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 76 , Issue 2 , April 2004 , pp. 235 - 246
Copyright
Copyright © Australian Mathematical Society 2004

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