Let $K$ be a closed, bounded and infinite subset of $\mathbb{C}$. We fix two non-zero natural integers $n$ and $m$. We introduce the $\mathbb{C}$-vector space $V = \mathbb{C}_n[X] \times \mathbb{C}_m[X]$ as well as the set: $$E = \left\{(Q,R) \in V \mid \|Q\|_K = \|R\|_K = 1\right\}.$$ Show that there exists a pair $\left(Q_0, R_0\right) \in E$ such that: $$\left\|Q_0 R_0\right\|_K = \inf\left\{\|QR\|_K \mid (Q,R) \in E\right\}.$$ To do this, one may equip $V$ with the norm defined by $$\|(Q,R)\| = \|Q\|_K + \|R\|_K$$ for $(Q,R) \in V$, then study the application $$\begin{aligned} f : E &\rightarrow \mathbb{R} \\ (Q,R) &\mapsto \|QR\|_K. \end{aligned}$$
Let $K$ be a closed, bounded and infinite subset of $\mathbb{C}$. We fix two non-zero natural integers $n$ and $m$. We introduce the $\mathbb{C}$-vector space $V = \mathbb{C}_n[X] \times \mathbb{C}_m[X]$ as well as the set:
$$E = \left\{(Q,R) \in V \mid \|Q\|_K = \|R\|_K = 1\right\}.$$
Show that there exists a pair $\left(Q_0, R_0\right) \in E$ such that:
$$\left\|Q_0 R_0\right\|_K = \inf\left\{\|QR\|_K \mid (Q,R) \in E\right\}.$$
To do this, one may equip $V$ with the norm defined by
$$\|(Q,R)\| = \|Q\|_K + \|R\|_K$$
for $(Q,R) \in V$, then study the application
$$\begin{aligned} f : E &\rightarrow \mathbb{R} \\ (Q,R) &\mapsto \|QR\|_K. \end{aligned}$$