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Suppose $X$ is normed space. Prove that $X^\ast$ is rotund iff $X/M$ smooth whenever $M$ is a closed subspace of $X$ such that $X/M$ is two-dimensional. I know that "a normed space is smooth iff each of its two-dimensional subspaces is smooth.", maybe that will be used.

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  • $\begingroup$ In case it helps anybody with the context of this question, this seems to be Exercise 5.40, p.492 from Megginson's book Introduction to Banach Space Theory. @Strongart: I think it is useful to mention context (in this case the book or lecture notes which you are studying), since it might help people answering your question to see what are you supposed to know already for the exercise. $\endgroup$ – Martin Sleziak Dec 26 '11 at 11:01

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