@DanielFischer I have also an other question.
We consider the linear mapping $T: \mathbb{R}^2 \to \mathbb{R}^2$ with $T(x_1, x_2)=(ax_1+bx_2, cx_1+dx_2), (x_1, x_2) \in \mathbb{R}^2$, where $a,b,c,d$ are given real numbers such that $|a|+|b|+|c|+|d| \neq 0$.
I want to show that the set $\{ M>0: ||T(x_1, x_2)||_2 \leq M ||(x_1, x_2)||_2 \forall (x_1, x_2) \in \mathbb{R}^2 \}$ is non-empty.

The following answer is given: $||T(x_1,x_2)||_2 \overset{\text{Cauchy-Scharz}}{\leq} \sqrt{a^2+b^2+c^2+d^2} ||(x_1,x_2)||_2 \forall (x_1,x_2) \in \mathbb{R}^2$.