Let for each $n$ there is a vector space $V_n$ given which carries two Banach-space norms $\|\cdot \|_i$ ($i=1,2$) which are
1) equivalent
2) $\|x\|_1\leqslant \|x\|_2$ for each $x\in V_n$.
Consider the $\ell_\infty$-sums: $X_i=\sum_{i=1}^\infty \oplus_\infty (V_n, \|\cdot\|_i)$ ($i=1,2$).
Must the spaces $X_1$ and $X_2$ be isomorphic?