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

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.