@DanielFischer Hey!!! Let $A: \mathbb{R}^n \to \mathbb{R}^n \\ A(x_1, x_2, \dots)=(a_1 x_1, a_2 x_2, \dots)$. In order to show that A is a linear mapping is the following sufficient?
Let $x=(x_1, x_2, \dots), y=(y_1, y_2, \dots) \in \ell^2(\mathbb{N}), \lambda, \mu \in \mathbb{R}$. $A(\lambda x+ \mu y)=(a_1(\lambda x_1+ \mu y_1), a_2 (\lambda x_2+ \mu y_2), \dots)=(\lambda a_1 x_1+ \mu a_1 y_1, \lambda a_2 x_2+ \mu a_2 x_2, \dots)=\lambda (a_1 x_1, a_2 x_2, \dots)+ \mu (a_1 y_1, a_2 y_2, \dots)= \lambda Ax+ \lambda Ay$