I am trying to work out an example my teacher gave in class and he skipped a few steps. We have $x= (x_1, x_2, x_3)$ Then we define a linear operator $T:\mathbb{R}^3 \rightarrow \mathbb{R}^3$ by $Tx = (x_2+x_3, 2x_1, x_2)$.
How do I work out that T is a bounded linear operator for this specific example?
First of all, actually every linear operator between finite-dimensional normed $\mathbb{R}$-vector spaces is bounded.
To get a specific bound in this example, equip $\mathbb{R}^3$ with the $\max$-norm $\|x\|=\max_i |x_i|$ (all norms are equivalent on finite-dimensional spaces, so we are free to choose) and note
$$\|Tx\|=\|(x_2+x_3,2x_1,x_2)\|\le 2 \|x\|$$
because $|x_2+x_3|, |2x_1|$ and $|x_2|$ are all $\le 2 \max_i |x_i|$.
