In the normed space $L^p[0,1] (1\leq p \leq \infty)$, linear operator V is defined as $(Vf)(x) = \int^x_0 f(t)dt , f\in L^p[0,1], x \in [0,1]$.
Then, show that $V$ is bounded linear operator from $L^p[0,1]$ to $L^p[0,1]$.
Indeed, $V$ is linear operator, since $V(af+bg)(x) = \int^x_0 (af+bg)(t)dt$. But, I am having some trouble showing "Bounded".
$\Vert V\Vert = sup \{\Vert Vf \Vert_p : \Vert f \Vert _p \leq 1,\ f \in L^p[0,1]\}$
How can I get $\Vert Vf \Vert_p$ is bounded from given conditions? Multiple integral and "^p" make this solution harder and harder. Could somebody help me?