# Immersion duals spaces

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 $$(\ell_\infty)'\subset (c_0)'$$?

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