Infinite norm of $T(x_1,\ldots,x_n) = \sum_{i = 1}^n a_ix_i$

Question

Let $\mathbb{K} = \mathbb{R},\mathbb{C}$. Compute the infinite norm of mapping $T:\mathbb{K}^n \to \mathbb{K}$ such that $T(x_1,\ldots,x_n) = \sum_{i = 1}^n a_ix_i$.

My notes show that $||T||_\infty = ||a||_1$. However, I have problems with the $\ge$ part. They state that I should take vector $x = (e^{i\theta_1},\ldots,e^{i\theta_n})$. Then I would have:

$|Tx| = |\sum e^{i\theta_i}a_i|$ it should be evident from here but I don't see it right now.

Answer 1

$|Tx| \le \sum_k |a|_k |x_k| \le \sum_k |a_k| \|x\|_\infty = \|a\|_1 \|x\|_\infty$.

Now choose $x_k = {\overline{a_k} \over |a_k|}$ (or one if $a_k = 0$), then $\|x\|_\infty = 1$ and $Tx = |\sum_k a_k {\overline{a_k} \over |a_k|}| = \sum_k |a_k| = \|a\|_1$