A note on the spaces of variable integrability and summability of Almeida and H\"ast\"o

We address an open problem posed recently by Almeida and H"ast"o in \cite{AlHa10}. They defined the spaces $\text{ellqp}$ of variable integrability and summability and showed that $\mathrm{\parallel }\cdot \mathrm{\mid }\text{ellqp}\mathrm{\parallel }$ is a norm if $q$ is constant almost everywhere or if ${\text{esssup}}_{x\in {\mathbb{R}}^n}1\mathrm{/}p(x)+1\mathrm{/}q(x)\le 1$. Nevertheless, the natural conjecture (expressed also in \cite{AlHa10}) is that the expression is a norm if $p(x),q(x)\ge 1$ almost everywhere. We show, that $\mathrm{\parallel }\cdot \mathrm{\mid }\text{ellqp}\mathrm{\parallel }$ is a norm, if $1\le q(x)\le p(x)$ for almost every $x\in {\mathbb{R}}^n\mathrm{.}$ Furthermore, we construct an example of $p(x)$ and $q(x)$ with $\min (p(x),q(x))\ge 1$ for every $x\in {\mathbb{R}}^n$ such that the triangle inequality does not hold for $\mathrm{\parallel }\cdot \mathrm{\mid }\text{ellqp}\mathrm{\parallel }$.