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If $X^*$ is separable space with norm then $X$ is separable. The opposite direction is not true since $\ell^1$ is separable but $\ell^\infty$ is not. Any conditions to make sure that if $X$ is separable then $X^*$ is also separable?

Arctic Char
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George Giatilis
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    https://math.stackexchange.com/questions/2388541/proving-that-a-banach-space-is-separable-if-its-dual-is-separable related – AlvinL Jul 26 '20 at 09:53
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    Sufficient is that $X$ is reflexive. – Jan Jul 26 '20 at 09:58
  • Some more questions that may help me understand some things about dual spaces.I)If #X# is reflexible then #X# is also reflexible;;Also is it true to say that:II)dimX<+00 if only and only if dimX<+00 and III)Any Banach space with dimX<+00 is reflexible? – George Giatilis Jul 26 '20 at 10:09
  • @GeorgeGiatilis This is not how this site works. Limit the post to one question (which is written in the body!), edit your post since it contains several (formatting) issues and find a suitable title for the question since it does not match what you are asking. Furthermore, include your own thoughts on the topic and finally, use Google, since I have seen the answers to all of your question on this site here some time ago. – Jan Jul 26 '20 at 10:13
  • This condition answers my question in a good way.I don't know if the opposite direction holds with a condition less general than reflexible spaces but it still helps.Thanks a lot! – George Giatilis Jul 26 '20 at 11:01

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