Let $x_1,...,x_n \in [0,1]$, $\lambda_1,...,\lambda_n \in \mathbb{C}$ and let $S,T : C([0,1]) \to C([0,1])$ be the operators defined by
$T(g)= \int_0^1 g(s)\, ds$
and
$S(g) = \sum_{i=1}^n \lambda_i g(x_i)$.
$C([0,1])$ is equipped with the $\|\cdot\|_{\infty}$-norm.
I already proved that $\|T\| = 1$ and $\|S\| = \sum_{i=1}^n |\lambda_i|$, where I used for the $\geq $-inequality the function $\tilde{g}$, which satisfies $\tilde{g}(x_i) = \dfrac{\bar{\lambda_i}}{|\lambda_i|}$ for all $i=1,...,n$ and is linearly interpolated between the $x_i$.
Now I want to find the norm of the operator $R:=T-S$. We obviously have
$\|R\| \leq \|T\|+\|S\| = 1+\sum_{i=1}^n |\lambda_i|$
and
$\|R\| \geq |\|T\|-\|S\|| = |1-\sum_{i=1}^n |\lambda_i||$.
How can I proceed now?
Thanks in advance!