I was trying to solve this problem:
Let $X = C[0,1]$ with $\|.\|_{\max}$ norm, let $\{ \alpha_{k}\}_{k=1}^{n}$ be real numbers, let $\{x_{k}\}_{k=1}^{n} \subset [0,1],$ and define $$ T f = \sum_{k=1}^{n} \alpha_{k} f(x_{k}). $$ Prove that $T$ is a bounded linear functional on $C[0,1]$ and find its norm.
My question is:
When I wrote the norm like this
$\|f\|_{\max} = \max_{\{x_{k}\}_{k=1}^{n} \in [0,1]} |f(x_{k})|$
My professor said that this way of writing is incorrect as I am taking the norm at specific points. So, what is the correct way of writing the norm in our problem?
Definition of the max norm is $\|f\|=\sup_{0 \leq x \leq 1} |f(x)|$.
Note that $|Tf| \leq \|f\| \sum\limits_{k=1}^{n}|\alpha_k|$ so $\|T\| \leq \sum\limits_{k=1}^{n}|\alpha_k|$. To show that equality golds think of a continuous function such that $f(x_k)$ is equal to the sign of $\alpha_k$ for each $k$.