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May 10, 2020 at 16:37 comment added Bill Johnson @TamaszKania: And a better one in our recent joint paper. :-)
May 10, 2020 at 15:39 comment added Tomasz Kania @BillJohnson, with the first known example being your space $J\! L$ :-)
May 10, 2020 at 13:59 comment added Bill Johnson Perhaps it is worth pointing out that $X$ has the property if and only if $X^*$ is weak$^*$ separable, which is a somewhat weaker condition than that $X$ embeds isomorphically into $\ell_\infty$.
May 10, 2020 at 13:15 vote accept Anupam
May 10, 2020 at 12:19 history edited Tomasz Kania CC BY-SA 4.0
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May 10, 2020 at 12:18 comment added YCor Thanks, I can see why I was stuck! In any case this provides useful context.
May 10, 2020 at 12:17 comment added Tomasz Kania @YCor, take $T\colon \ell_\infty\to c_0$ given by $T (\xi_k)_{k=1}^\infty = (\xi_k / k)_{k=1}^\infty$. It is compact and injective.
May 10, 2020 at 12:16 history edited Tomasz Kania CC BY-SA 4.0
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May 10, 2020 at 12:16 comment added YCor I was actually trying to check that it never exists if $X$ is non-separable.
May 10, 2020 at 12:14 history answered Tomasz Kania CC BY-SA 4.0