The title says it all. Does there exist a (separable, infinite-dimensional) Banach space $E$ with the normed space $L(E)$ of bounded, linear operators thereon separable?
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6$\begingroup$ Argyros & Haydon gave an example of an infinite-dimensional Banach spaces $E$ such that every operator is of the form compact + scalar. Would that imply that $L(E)$ is separable? doi.org/10.1007/s11511-011-0058-y $\endgroup$– Gerald EdgarCommented Jul 15, 2018 at 18:52
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$\begingroup$ @GeraldEdgar Why the question mark? $\endgroup$– Yemon ChoiCommented Jul 15, 2018 at 19:15
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1$\begingroup$ @YemonChoi ... Because Parschalien should think about it. This is also why it was a comment and not an answer. $\endgroup$– Gerald EdgarCommented Jul 15, 2018 at 19:17
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