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



Strictly convex equivalent norm

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> 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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[Link.](http://books.google.co.uk/books?id=1A7ppEZPXEIC&pg=PA33&lpg=PA33&dq=equivalent+strictly+convex+norm&source=bl&ots=zx13lvLJXV&sig=bBaKVMIB4be6BpBQlUcrpQaTEUQ&hl=en&ei=aaMvTM2jH9S5jAeY843DBQ&sa=X&oi=book_result&ct=result&resnum=5&ved=0CC8Q6AEwBA#v=onepage&q=equivalent%20strictly%20convex%20norm&f=false)


