Let $V = C^1([-1,1],\mathbb{R})$, the set of continuously differentiable functions on $[-1,1]$. $V$ is equipped with the norm $||f||_1 = \sup_{x\in[-1,1]} |f(x)|$.
Define a linear continuous map $L:V\rightarrow \mathbb{R}$ by $L(f) = \int_{-1}^{1} \sin(\pi t)f(t) dt$.
I am trying to find the value of the $||L||_{V'}$, the norm in the dual space.
I have found that $|L(f)| \leq 2 ||f||_1$for all $f\in V$. So I expect that $||L||_{V'} = 2$. However I have not been able to find a function that satisfies $$\sup_{||f||_1 = 1} |L(f)| = 2$$ which would give me the answer. Any suggestions?