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The classical Banach spaces $\ell _{ \varphi }/h_{ \varphi }$


Authors: Antonio S. Granero and Henryk Hudzik
Journal: Proc. Amer. Math. Soc. 124 (1996), 3777-3787
MSC (1991): Primary 46B20
DOI: https://doi.org/10.1090/S0002-9939-96-03490-9
MathSciNet review: 1343694
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Abstract: In this paper we study some structural and geometric properties of the quotient Banach spaces $ \ell _{\varphi }(I)/ h_{\varphi }(\mathcal {S})$, where $I$ is an arbitrary set, $ \varphi $ is an Orlicz function, $ \ell _{\varphi }(I)$ is the corresponding Orlicz space on $I$ and $ h_{\varphi }(\mathcal {S}) =\{x\in \ell _{\varphi }(I) :\forall \lambda >0,\ \exists s\in \mathcal {S}\text { such that } I_{\varphi } (\frac {x-s}{\lambda })<\infty \}$, $ \mathcal {S}$ being the ideal of elements with finite support. The results we obtain here extend and complete the ones obtained by Leonard and Whitfield (Rocky Mountain J. Math. 13 (1983), 531-539). We show that $ \ell _{\varphi }(I) / h_{\varphi }(\mathcal {S})$ is not a dual space, that $Ext(B_{ \ell _{\varphi }(I)/ h_{\varphi }(\mathcal {S}) })=\emptyset $, if $ \varphi (t)>0$ for every $t>0$, that $S_{ \ell _{\varphi }(I)/ h_{\varphi }(\mathcal {S})}$ has no smooth points, that it cannot be renormed equivalently with a strictly convex or smooth norm, that $ \ell _{ \varphi }(I)/h_{ \varphi }(\mathcal {S})$ is a Grothendieck space, etc.


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Additional Information

Antonio S. Granero
Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040-Madrid, Spain

Henryk Hudzik
Affiliation: Faculty of Mathematics and Computer Science, A. Mickiewicz University, Poznań, Poland
Email: hudzik@plpuam11.bitnet

DOI: https://doi.org/10.1090/S0002-9939-96-03490-9
Keywords: Orlicz spaces, quotient spaces
Received by editor(s): March 15, 1995
Received by editor(s) in revised form: June 13, 1995
Additional Notes: The first author was supported in part by DGICYT grant PB 94-0243. The paper was written while the second author visited the Universidad Complutense de Madrid.
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1996 American Mathematical Society

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