I have a few questions about dual spaces of Banach spaces and what does it mean to two spaces to be equall in this context.
I have a note in my notebook that says "If a space $Y$ is the dual space of other space $X$ then $\overline{B(0,1)} \subset Y$ is $w^*$-compact on $Y$ by Banach-Alaoglu "
The example I have is $\ell^1$: "$\ell^1$ is the dual of $c_0$, and hence the unit ball of $\ell^1$ is $w^*$-compact".
My problem here is what does it mean the word "is"? What class of isomorphism is referring?
I know that $\ell^1$ is isometrically isomorphism to $c_0^*$, and because of this I can make a reasoning like this: By banach alaoglu the unit ball of $c_0^*$ is $w^*-$compact, now consider the $w^*$ topology on $\ell^1$. What I think this means is to put a topology on $\ell^1$ via the identification we have between $\ell^1$ and $c_0^*$, now since this identificacion is an isomorphism, i.e preserves the norm, then the unit ball of $c_0^*$ is identifyed with the unit ball of $\ell^1$ and hence the unit ball of $\ell^1$ is $w^*-$compact. But if the identification is not an isometry I am not sure if I can conclude that the unit ball of $\ell^1$ is $w^*-$compact.