Let $X$ be a separable Banach space. Show that there is a surjective linear operator $P: L^1(\mathbb N) \to X$ and a subspace $S \subset L^1(\mathbb N)$ such that $$\|P(x)\| = \inf_{y \in S} \|x+y\|.$$
Update: Consider a dense sequence $s_n$ in $X$ and define $P$ as follows. $$P(x) : = x_1s_1 + \dots + x_ns_n + \dots.$$ It is true that $P$ is onto due to density and absolute convergence. It is left to show the norm result. First, show that $$\|P(x)\| \leq \inf_{y \in S} \|x+y\|.$$ That is, $$\|P(x)\| = \|x_1s_1 + \dots + x_ns_n + \dots\| \leq \|x+y\|, \ \forall y \in S.$$ How to proceed from here? Also how to prove the other direction, i.e., $$\|P(x)\| \geq \inf_{y \in S} \|x+y\|?$$
Hint: let $s_i, i \in \mathbb N$ be a dense sequence in the unit ball of $X$, and $P (x) = \sum_i x_i s_i$.