The approximation property for spaces of holomorphic functions on infinite dimensional spaces II
Let H ( U ) denote the vector space of all complex-valued holomorphic functions on an open subset U of a Banach space E. Let τ ω and τ δ respectively denote the compact-ported topology and the bornological topology on H ( U ) . We show that if E is a Banach space with a shrinking Schauder basis, and...
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| Vydáno v: | Journal of functional analysis Ročník 259; číslo 2; s. 545 - 560 |
|---|---|
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| Jazyk: | angličtina |
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Elsevier Inc
01.07.2010
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| ISSN: | 0022-1236, 1096-0783 |
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| Abstract | Let
H
(
U
)
denote the vector space of all complex-valued holomorphic functions on an open subset
U of a Banach space
E. Let
τ
ω
and
τ
δ
respectively denote the compact-ported topology and the bornological topology on
H
(
U
)
. We show that if
E is a Banach space with a shrinking Schauder basis, and with the property that every continuous polynomial on
E is weakly continuous on bounded sets, then
(
H
(
U
)
,
τ
ω
)
and
(
H
(
U
)
,
τ
δ
)
have the approximation property for every open subset
U of
E. The classical space
c
0
, the original Tsirelson space
T
∗
and the Tsirelson
∗–James space
T
J
∗
are examples of Banach spaces which satisfy the hypotheses of our main result. Our results are actually valid for Riemann domains. |
|---|---|
| AbstractList | Let
H
(
U
)
denote the vector space of all complex-valued holomorphic functions on an open subset
U of a Banach space
E. Let
τ
ω
and
τ
δ
respectively denote the compact-ported topology and the bornological topology on
H
(
U
)
. We show that if
E is a Banach space with a shrinking Schauder basis, and with the property that every continuous polynomial on
E is weakly continuous on bounded sets, then
(
H
(
U
)
,
τ
ω
)
and
(
H
(
U
)
,
τ
δ
)
have the approximation property for every open subset
U of
E. The classical space
c
0
, the original Tsirelson space
T
∗
and the Tsirelson
∗–James space
T
J
∗
are examples of Banach spaces which satisfy the hypotheses of our main result. Our results are actually valid for Riemann domains. |
| Author | Mujica, Jorge Dineen, Seán |
| Author_xml | – sequence: 1 givenname: Seán surname: Dineen fullname: Dineen, Seán email: sean.dineen@ucd.ie organization: Department of Mathematics, University College Dublin, Belfield, Dublin 4, Ireland – sequence: 2 givenname: Jorge surname: Mujica fullname: Mujica, Jorge email: mujica@ime.unicamp.br organization: Departamento de Matemática, Universidade Estadual de Campinas, Rua Sergio Buarque de Holanda 651, 13083-859 Campinas, SP, Brazil |
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| Cites_doi | 10.4064/sm-122-2-139-151 10.5802/aif.419 10.1016/j.jmaa.2006.05.076 10.1007/s13163-009-0026-7 10.1007/BF01419608 10.1016/j.jat.2004.01.008 10.1016/S1385-7258(67)50051-3 10.5802/aif.68 10.1016/0022-1236(76)90026-4 10.1007/BF01351314 10.1016/S0019-3577(01)80022-7 10.1016/0022-1236(83)90081-2 10.1007/BF01078599 10.1090/S0002-9939-1984-0728358-5 10.4064/sm-16-2-173-182 10.1007/BF02385668 10.5802/aif.345 10.1007/BF02392870 10.1216/rmjm/1181071856 |
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| Keywords | Pseudoconvex Riemann domain Holomorphic function Banach space Schauder basis |
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| Snippet | Let
H
(
U
)
denote the vector space of all complex-valued holomorphic functions on an open subset
U of a Banach space
E. Let
τ
ω
and
τ
δ
respectively denote... |
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| SubjectTerms | Banach space Holomorphic function Pseudoconvex Riemann domain Schauder basis |
| Title | The approximation property for spaces of holomorphic functions on infinite dimensional spaces II |
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