Norm of $ T:(C([0,1]),||\cdot||_\infty)\rightarrow\mathbb{R}$ where $Tf=\sum_{k=1}^n a_kf(t_k)$? I need to calculate the operator Norm of the linear operator defined as:
$$ T:(C([0,1]),||\cdot||_\infty)\rightarrow\mathbb{R} \text{ where } Tf=\sum_{k=1}^n a_kf(t_k)$$
for $0\leq t_1<t_2<...<t_n\leq 1$ and $a_1,...,a_n\in \mathbb{R}$.
I have been able to show that $||T||\geq\left|\sum_{k=1}^n a_k\right|$ and $||T||\leq \sum_{k=1}^n|a_k|$ but I don't seem to able to bound it further than that. Would appreciate any help. Thank you.
 A: WLOG, one can assume that $a_k\ne0$ for all $k$ (if not, just drop all the zero-values, and in the case that all are zero, the problem is trivial).
Take $f(x)$ a continuous function with these properties:

*

*$f(t_k)=\frac{|a_k|}{a_k}$.

*$|f(x)|\le 1$ for all $x\in[0,1]$.

For example, you can take the polygonal through the points $\{(t_k,f(t_k)),\,1\le k\le n\}\cup\{(s_k,0),1\le k\le n-1\}$, where $s_k=\frac{t_k+t_{k+1}}{2}$ (the mid point of the interval $[t_k,t_{k+1}]$).
Then it is clear that $\|f\|_\infty=1$ (observe that $|f(t_k)|=1$) and $Tf=\sum_{k=1}^n |a_k|$.
This together with your calculations leads to $\|T\|=\sum_{k=1}^n|a_k|$.
A: It might help to separate $T$ into the composition of two simpler maps.
Note that the map $s:C[0,1] \to \mathbb{R}^n$ defined by
$s(f) = (f(t_1),...,f(t_n))$ has norm one (using the $l_\infty$ norm on $\mathbb{R}^n$) and is surjective.
Furthermore, for any $y$ with $\|y\|_\infty =1$ it is easy to construct (by interpolation & extrapolation) some $f \in C[0,1]$ with $\|f\| = 1$ such that $s(f) = y$.
Let the operator $\tau:\mathbb{R}^n \to \mathbb{R}$ be given by $\tau(y) = \sum_k a_k y_k$. It is straightforward to show that $\|\tau\| = \|a\|_1$ and there is some
$y \in \mathbb{R}^n$ such that $\|y\|_\infty = 1$ and $\tau(y) = \|a\|_1$.
Note that $T = \tau \circ s$.
Hence $\|T\| \le \|s\| \|\tau\| = \|a\|_1$.
Suppose $\tau(y) = \|a\|_1$ with $\|y\|_\infty = 1$ and $s(f) = y$ where $\|f\| = 1$, then
$Tf = \tau(s(f))  = \tau(y) = \|a\|_1$ and so $\|T\| = \|a\|_1$.
