Consider the map $\phi : \ell^1 \to \prod_{y \in \ell^\infty} \mathbb R_y$, $x \mapsto (\langle y,x\rangle_{\ell^\infty,\ell^1})_{y \in \ell^\infty}$, where $\langle y,x\rangle_{\ell^\infty,\ell^1}= \sum_{n \ge 1} y_n x_n$ and denote by $K_1(0,\ell^1)$ the closed unit ball in $\ell^1$.
I'm trying to show that $\phi(K_1(0,\ell^1)) \subset \prod_{y \in \ell^\infty}[-\|y\|_\infty,\|y\|_\infty]$ and that the closure of $\phi(K_1(0,\ell^1))$ is contained in $\prod_{y \in \ell^\infty}[-\|y\|_\infty,\|y\|_\infty]$ with respect to the product topology.
But I'm kind of lost and I'm not sure where to start to show these statements. I would be glad if someone could help me and give me some hint.