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It is known that in a reflexive Banach space, if the norm is strictly convex, then its dual will be smooth Banach space, and if the norm is smooth, then the dual norm is strictly convex.

We can find an equivalent norm on $\mathbb{R}^2$ such that the norm is not strictly convex; however, we can see that the space is smooth. However, I am still looking for an infinite-dimensional reflexive Banach space such that the space is smooth, but its norm is not strictly convex. I appreciate any help you can provide.

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    $\begingroup$ Start from a smooth but not strictly convex norm on $\mathbf{R}^2$ and take the $\ell^2$-direct sum with an infinite-dimensional Hilbert space? Well this answers the question in the body but not the question in the title. Please fix your question so that the question in the body coincides with that in the title. $\endgroup$
    – YCor
    Commented Apr 26, 2023 at 5:39
  • $\begingroup$ Thank you so much. However, I am also looking for an $l_2$ space with an equivalent norm such that the norm is smooth but not strictly convex. $\endgroup$
    – PPB
    Commented Apr 26, 2023 at 7:56
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    $\begingroup$ I may miss something, but doesn't the example in my comment solve your question? $\endgroup$
    – YCor
    Commented Apr 26, 2023 at 8:39

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