In this paper of Zubelevich the author consider a a scale of Banach spaces $\{(E_{s},\|\cdot\|_{s}):0<s<1\}$, that is to say, each $(E_{s},\|\cdot\|_{s})$ is a Banach space and
$$ E_{s+\delta}\subseteq E_{s},\quad \|\cdot\|_{s}\leq \|\cdot\|_{s+\delta}, \quad s+\delta<1, \delta>0, $$ for each $0<s<1$. Then, fixed $a>1$, defines the (open) triangle $$ \Delta :=\{(\tau,s)\in\mathbb{R}^{2}:\tau>0,0<s<1,1-s-a\tau>0 \}, $$ and the seminormed space $$ E:=\bigcap_{(s,\tau)\in \Delta} C([0,\tau],E_{s}), $$ endowed the family of norms $\|u\|_{\tau,s}:=\max_{0\leq t\leq \tau}\|u(t)\|_{s}$. As usual, $C([0,\tau],E_{s})$ denotes the space of the continuous maps $u:[0,\tau]\longrightarrow (E_{s},\|\cdot\|_{s})$. So, $E$ is a (locally convex) topological space with a basis of the topology given by the open balls $B_{\tau,s}:=\{u\in E: \|u\|_{\tau,s}<r\}$, for each $r>0$.
My question is the following: Is $E$ non-empty? Or, more precisely, under that conditions $E$ is not a "trivial" (for instance, a single point) space?
Thank you very much in advance for your comments.