Abstract.
We introduce the classes of locally convex spaces with the local Dunford-Pettis property and locally dual Schur spaces. We examine their properties and their relationship to other classes of locally convex spaces. In the class of locally convex spaces with the local Dunford-Pettis property all polynomials are weakly sequentially continuous whereas in the class of locally dual Schur spaces all polynomials are weakly continuous on bounded sets.
Similar content being viewed by others
References
K-D Bierstedt RG Meise WH Summers (1982) Köthe sets and Köthe sequence spaces JA Barross (Eds) Functional Analysis, Holomorphy and approximation theory North-Holland Amsterdam 27–91
Bombal F (1989) Operators on vector sequence spaces. In: Martin-Peinador E, Rodés A (eds) Geometric Aspects of Banach Spaces. LMS Lect Notes 140: 94–106. Cambridge: Univ Press
C Boyd RA Ryan (1998) ArticleTitleBounded weak continuity of homogeneous polynomial at the origin Arch Math 71 211–218 Occurrence Handle0922.46041 Occurrence Handle10.1007/s000130050254 Occurrence Handle1637369
P Perez Carreras J Bonet (1987) Barrelled Locally Convex Spaces North-Holland Amsterdam Occurrence Handle0614.46001
Cembranos P (1982) Algunas propiedares del espacio de Banach C(K,E). Thesis, Universidad Complutense Madrid
JM Castillo R García R Gonzalo (1999) ArticleTitleBanach spaces in which all multilinear forms are weakly sequentially continuous Studia Math 136 121–145 Occurrence Handle0948.46010 Occurrence Handle1716170
Diestel J (1980) A survey of results related to the Dunford-Pettis property. In: Graves WH (ed) Integration, Topology and Geometry in Linear Spaces, pp 15–60. Providence, RI: Amer Math Soc
JC Díaz (1989) ArticleTitleMontel subspaces in the countable projective limits of L p(μ)-spaces Can Math Bull 32 169–176 Occurrence Handle0645.46004
S Dineen (1999) Complex Analysis on Infinite Dimensional Spaces Springer Berlin Heidelberg New York Occurrence Handle1034.46504
Floret K (1980) Weakly Compact Sets. Lect Notes Math 801. Berlin Heidelberg New York: Springer
A Grothendieck (1953) ArticleTitleSur les applications lineares faiblement compactes d’espaces du type C(K) Can J Math 5 129–173 Occurrence Handle0050.10902 Occurrence Handle58866
J Horváth (1966) Topological Vector Spaces and Distributions NumberInSeries1 Addison-Wesley Reading, Mass Occurrence Handle0143.15101
H Jarchow (1981) Locally Convex Spaces Teubner Stuttgart Occurrence Handle0466.46001
H Junek (1977) ArticleTitleOn dual spaces of locally convex spaces defined by ideals Serdica 3 227–235 Occurrence Handle0348.46003 Occurrence Handle625150
H Junek (1983) Locally Convex Spaces and Operator Ideals Teubner Leipzig Occurrence Handle0552.46005
MA Miñarro (1995) ArticleTitleA characterization of quasinormable Köthe sequence spaces Proc Amer Math Soc 123 1207–1212 Occurrence Handle0832.46001 Occurrence Handle10.2307/2160720 Occurrence Handle1227526
H Nues (1978) ArticleTitleUber die Regularitätsbegriffe induktiver lokalkonvexer Sequenzen Manuscripta Math 25 135–145 Occurrence Handle10.1007/BF01168605 Occurrence Handle482036
A Pietsch (1980) Operator Ideals North-Holland Amsterdam Occurrence Handle0434.47030
VS Retakh (1970) ArticleTitleSubspaces if a countable inductive limit Soviet Math Dokl 11 1384–1386 Occurrence Handle0213.12504
HP Rosenthal (1978) ArticleTitleSome recent discoveries in the isomorphic theory of Banach spaces Bull Amer Math Soc 84 803–831 Occurrence Handle0391.46016 Occurrence Handle499730 Occurrence Handle10.1090/S0002-9904-1978-14521-2
M Valdivia (1982) Topics in Locally Convex Spaces North-Holland Amsterdam Occurrence Handle0489.46001
M Venkova (2004) ArticleTitleProperties of Q-reflexive locally convex spaces J Korean Math Soc 41 51–64 Occurrence Handle1053.46027 Occurrence Handle2048700 Occurrence Handle10.4134/JKMS.2004.41.1.051
L Weis (1978) ArticleTitleÜber schwach folgenpräkompakte Operatoren Arch Math 30 411–417 Occurrence Handle0424.47016 Occurrence Handle10.1007/BF01226076 Occurrence Handle636122
Author information
Authors and Affiliations
Additional information
Research supported by Science Foundation Ireland, Basic Research Grant 2004.
Rights and permissions
About this article
Cite this article
Boyd, C., Venkova, M. Grothendieck space ideals and weak continuity of polynomials on locally convex spaces. Mh Math 151, 189–200 (2007). https://doi.org/10.1007/s00605-007-0483-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-007-0483-3