Question

Does any Banach space admit an equivalent strictly convex norm (i.e. such a norm, that a unit sphere does not contain segments)?

Answer 1

Every separable Banach space has an equivalent norm which is both strictly convex and smooth. For certain nonseparable spaces, in particular, $\ell_{\infty}(\Gamma)$ with $\Gamma$ uncountable, there may be no equivalent strictly convex or smooth norm.

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