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Is correct say that $(\ell_\infty)'\subset (c_0)'$?

My idea is the following:

I know that $c_0\subset\ell_\infty$. Then, the function $$ \begin{split} \Psi:(\ell_\infty)'&\to(c_0)'\\ f&\mapsto\Psi(f)=f|_{c_0} \end{split} $$ is injective.

Is correct the function? Does exist an injective function $\Psi$ such that $(\ell_\infty)'\subset (c_0)'$?

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  • $\begingroup$ what is $c_0$ in your question? $\endgroup$ – Pink Panther Apr 8 at 6:51
  • $\begingroup$ $\ell_\infty^*$'s cardinality exceeds that of the continuum; the same can't be said for $c_0^*=\ell_1$'s cardinality. $\endgroup$ – David Mitra Apr 8 at 11:10
  • $\begingroup$ The restriction map you define is NOT injective! This is not completely obvious but follows from the Hahn-Banach theorem. $\endgroup$ – Jochen Apr 9 at 7:47
  • $\begingroup$ Thank you so much! $\endgroup$ – Jose Esparrago Apr 30 at 7:10

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