# functional analysis – Calculate \$ | T | \$ when \$ D (T) \$ has \$ | | _ | {{infty} \$ norm

Let $$X = {f in C[0,1]: f (0) = 0 }$$. To define $$T: X to mathbb R$$
$$T_f = int_0 ^ 1 f (t) dt$$
Calculate $$| T |$$ when $$X$$ is equipped with $$| | _ { infty}$$.

By getting closer, I've $$| T_f | leq int_0 ^ 1 | f (t) | dt leq sup_ {t in [0,1]} | f (t) | int_0 ^ 1 dt = || f || _ { infty}$$
So, $$| T_f | 1$$.

My challenge is to show $$| T_f | geq 1$$. To do this, I have to find a $$f in X$$ such as $$| f | 1$$ and $$| T_f | = 1$$. Someone can help. Thank you