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