I am sorry if the question is easy but can one give me an example of a pair of Banach spaces, say $X$ and $Y$, $X$ isomorphic to $Y$ such that $X$ has no isometric copy of $Y$ neither $Y$ has isometric copy of $X$ inside?
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closed as off topic by Benoît Kloeckner, Loop Space, Andreas Thom, Bill Johnson, Ryan Budney Jul 30 '11 at 8:38Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here.If this question can be reworded to fit the rules in the help center, please edit the question. 


Space $X = l_1$ is separable, therefore has an equivalent norm which is strictly convex. Let $Y$ be the space with that norm. Now every subspace of $Y$ is strictly convex, and so it remains to show that for any $2$dimensional subspace of $X$, there is a line segment in the unit sphere. 

