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

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.