I know that $c_{00}$ is dense in $c_0$ in $l_1$. Is it also true that $c_{00}^*$ dense in $c_0^* $?
What about in $l_\infty$? I know that it's not true that $c_{00}$ is dense in $c_0$ in $l_\infty$. But I saw it somewhere that $c_{00}$ is dense in $c_0$ in $l_\infty$.
Why is this true?
Thanks!
Of course $c_{00}$ is dense in $c_0$ in the infinity norm. If $(x_n)\in c_0$ and $\varepsilon>0$, then we may find $n_0$ s.t. for all $n\geq n_0$ we have $|x_n|<\varepsilon$. Set $y=(x_1,\dots,x_{n_0},0,0,\dots)\in c_{00}$. Then $\|x-y\|_{\infty}=\sup_{n\geq n_0}|x_n|\leq\varepsilon$.
Now about the duals. If $X$ is a normed space and $Y$ is a subspace of $X$, then $Y^*\subset X^*$: This is not trivial! This makes sense by the Hahn-Banach theorem: any functional on a subspace can be extended to a functional of the same norm defined on the entire space. Now if $Y$ is dense in $X$, then this extension is unique (why?). This shows that $Y^*\cong X^*$, so to answer your question, since $c_{00}$ is dense in $c_0$, we have that $c_{00}^*$ is dense in $c_0^*$, since they are isometrically isomorphic.