Define the functional $L:C([0,1], \mathbb{R}) \rightarrow \mathbb{R}$ by $L(\varphi) = \displaystyle\int\_0^{1/2}\varphi(t)dt - \displaystyle\int\_{1/2}^{1}\varphi(t)dt$. we consider in $C([0,1], \mathbb{R})$ the supremum norm. I have showed that $||L|| = 1$, however I couldn't show that there is no $\varphi \in C([0,1], \mathbb{R})$ such that $||\varphi|| = |L(\varphi)|$. 

This exercise was proposed before the Hanh-Banach Theorem, so I don't know if you need this Theorem. Does anyone have any idea how to do?