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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?

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    Let each $V_n=\Bbb R$. For $x\in V_n$ let $\|x\|_1=|x|$ and $\|x\|_2=n|x|$. Then $X_1$ is just $\ell_\infty$; is $X_2$?2012-10-07
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    @TMK The summation index should be $n$, not $i$, no?2012-10-07
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    Is it true that norms $\Vert\cdot\Vert_1$, $\Vert\cdot\Vert_2$ depends on $n$. Otherwise you had to state that all $V_n$ are the same2012-10-07
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    @BrianM.Scott: the map $x\mapsto n^{-1}x$ is a linear isometry of $(\mathbb{R},\Vert \cdot \Vert_1)$ onto $(\mathbb{R},\Vert \cdot \Vert_2)$, so in the case you mention $X_2$ *is* also $\ell_\infty$.2012-10-12

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