Here is the link of the function I am speaking about:
Finding the norm of an operator.
And here is the function:
$$ f(x) = \begin{cases} 1 & x = \alpha_k \geq 0 \\ -1 & x = \alpha_k < 0 \\ 0 & x \in \{0,1\} \setminus \{\alpha_k\}_{k =1}^n \end{cases} $$
Here is the question:
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 questions are:
(1) In the defined function, should not we respect the given that "$\{x_{k}\}_{k=1}^{n} \subset [0,1]$ " How can we say that $x = \alpha_{k} \geq 0 $? is not the $\{ \alpha_{k}\}_{k=1}^{n}$ are real numbers?
(2) How can I prove that the defined function continuous?