Looking for a problem, I find one inequality difficult to proove. I first present the data of my problem as follows:
For all subgroup $G$ of $(\Bbb C,+)$, we define:$$\delta(G)=\inf\limits_{g \in G, g \neq 0} |g|$$ For all $\Bbb R-$linear map $f$ from $\Bbb C$ to $\Bbb C$ as $\Bbb R-$ vectorial space, we define: $$\|f\|=\sup\limits_{z\in \Bbb C, z \neq 0} \frac{|f(z)|}{|z|}$$
Let $G$ be a subgroup of $(\Bbb C,+)$ such that there exists $(e_1,e_2) \in G^2$ s.t. $(e_1,e_2)$ is a basis of the $\Bbb R$ vectorial space $\Bbb C$. Let $\phi$ the linear map from $\Bbb C$ to $\Bbb C$ s.t. , for $z=x+yi$ where $(x,y) \in \Bbb R^2$ we have $\phi(z)=xe_1+ye_2$.
1) Prove that $\Bbb Z^2 \subset H$.
2) Prove that $$\delta(H)\cdot\|\phi^{-1}\| \leq \delta(G) \leq \delta (H). \|\phi\|.$$
I answered all questions except the inequality $$\delta(H)\cdot\|\phi^{-1}\| \leq \delta(G)$$
about which I need your help.
Thank you very much in advance.