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?
closed as off topic by Benoît Kloeckner, Andrew Stacey, Andreas Thom, Bill Johnson, Ryan Budney Jul 30 '11 at 8:38
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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.