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Let $X$ be a Banach space and $X_1\xrightarrow{}X$ isometrically. Under some assumption can we guarantee that $X^*$ contains an isometric copy of $X_1^*$. I am also interested to know if this happens for particular examples for example if $X_1=\ell_\infty^n,\ell_1^n$ or $\ell_2^n.$

I know we can have something like this of $X$ is $K$-convex and $X_1=\ell_2^n.$ This is because the dual of $K$-convex Banach space is again $K$-convex.

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    $\begingroup$ That $X_1$ is 1-complemented in $X$ will do. $\endgroup$
    – Tomasz Kania
    Apr 27, 2021 at 13:36
  • $\begingroup$ @ Tomasz. Thanks for this. What will be the short proof of this? So by your observation if $X_1$ is $\ell_\infty$ then we have that what we want. Do we also have that $X_1^*$ is complemented again? $\endgroup$
    – A beginner mathmatician
    Apr 27, 2021 at 14:04

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