evinda

@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 $x=(x_1, x_2, \dots), y=(y_1, y_2, \dots) \in \ell^2(\mathbb{N}), \lambda, \mu \in \mathbb{R}$. $A(\lambda x+ \mu y)=(a_1(\lambda x_1+ \mu y_1), a_2 (\lambda x_2+ \mu y_2), \dots)=(\lambda a_1 x_1+ \mu a_1 y_1, \lambda a_2 x_2+ \mu a_2 x_2, \dots)=\lambda (a_1 x_1, a_2 x_2, \dots)+ \mu (a_1 y_1, a_2 y_2, \dots)= \lambda Ax+ \lambda Ay$

Hello @DanielFischer !!!
I am looking at the following:
The nodes $A,B,C,D,E,F,G,H,I$ are given and are related as follows:
http://www.directupload.net/file/d/4209/fiinaajz_png.htm
The nodes $B,C,D,E,F,G,H,I$ are tasks that are done so that a work is completed , for example a building, and the cost $p(a,b)$ are the days that are needed so that the work $b$ finishes if the work $a$ has already finished.
For example, $p(A,B)=22$: 22 days are needed so that the work B is done.
$p(B,E)$: 16 days are needed so that the work E is done if the work B has finished.

@DanielFischer We have the linear mapping $T: \ell^2 (\mathbb{N}) \to \ell^2(\mathbb{N})$ with $T(x_1, x_2, x_3, \dots)=(x_2, x_3, \dots)$
I have shown that $||T^n||=1 \forall n=1,2, \dots$ as follows:

We notice that:
$T^2(x_1, x_2, \dots)=(x_2, x_3, \dots), T^3(x_1, x_2, \dots)=(x_3, x_4, \dots), \dots, T^n(x_1, x_2, \dots)=(x_n, x_{n+1}, \dots)$

Since $x \in \ell^2(\mathbb{N}), ||T^n||= \sup \{ ||T^n x||: ||x||_2=1\}=1$

But then I have to show that $||T^n x||_2 \to 0 \forall x \in \ell^2(\mathbb{N})$.

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