Let $V=C([-1,1],\mathbb R)$ be the space of continuous functions from $[-1, 1]$ to $\mathbb R$ equipped with the norm $\|f\|_{\infty}=\sup_{x\in\mathbb R}|f(x)|$. Consider the mapping $L:V\to \mathbb R$ defined by $$L(f):=\int_{-1}^1\sin(\pi t)f(t)\mathrm d t\qquad\forall f\in V.$$ Show that $L\in V'$ and find its norm $\|L\|_{V'}$.
In the above, $V'$ refers to the dual space of $V$. I struggled with this question. Firstly, I wasn't sure how $\|\cdot\|_{V'}$ is defined. Is it the supremum norm of $V'$? Secondly, when I tried to visualize $\sup_{f\in V}|L(f)|$, I thought of $f$ as being a function with a steep gradient, with $f(0)=0$, $f(t)>0$ when $t>0$ and $f(t)<0$ when $t<0$ so that $\sin(\pi t)f(t)$ is positive. The steeper the gradient of $f$, the greater the area underneath $\sin(\pi t)f(t)$, without limit. So I could not find $\sup_{f\in V}|L(f)|$.
In a nutshell, how am I supposed to know how the norm $\|\cdot\|_{V'}$ is defined?
Edit: Why was I required to show that $L\in V'$? I ignored that because I thought it was obvious, but another question in the tutorial asked the same thing. How does one "show" that?
As $V$ is a normed vector space, we associate to each $L \in V'$, the number $\left\| L \right\| = \sup\limits_{x\in V, \left\|x\right\| \le 1} L(x)$.
This number is called the norm of $L$ in $V'$, as it makes $V'$ a normed space.
P.S., one can check that $L\in V'$ by verifying the linearity of the addition and scalar multiplication.