Check if the functional is continuous Check if following funtional is continuous 
$ P\ni p \to q \in P : q'=p ,  q(0)=0$ where $ P $ is polynomials space
towards norms:
1) $||p|| = sup{|p(t)|: t ∈ [0, 1]}$
2)$||p|| = \int\limits_{0}^{1}{|p(t)|: t ∈ [0, 1]}$
if so, compute its norm.
I've started:
1)$||p|| = sup_{sup{|p(t)|\le1}}\{sup{|q(t)|: t ∈ [0, 1]}\}$
2)
$ ||p|| = sup_{\int\limits_{0}^{1}{|p(t)|\le1}}\{\int\limits_{0}^{1}{|q(t)|: t ∈ [0, 1]}\} $
But I don't know how to continue. Could I ask for help?
 A: Let's call $\varphi$ the functional and $\Vert p \Vert_\infty$ the norm $\sup\limits_{t \in [0,1]} \vert p(t) \vert$.
For (1), you have for all $t \in [0,1]$
$$[\varphi(p)](t) = \int_0^t p(s) \ ds$$ hence
$$\vert [\varphi(p)](t) \vert = \left\vert \int_0^t p(s) \ ds \right\vert \le \int_0^t \Vert p \Vert_\infty \ ds = t \Vert p \Vert_\infty \le \Vert p \Vert_\infty.$$ Therefore
$$\Vert \varphi(p) \Vert_\infty \le \Vert p \Vert_\infty$$ which proves the continuity of $\varphi$ for the norm $\Vert \cdot \Vert_\infty$.
Taking for $p$ the polynomial function $p_1$ which is constant and equal to $1$ you have $\Vert p_ 1 \Vert_\infty = 1$ and $\Vert \varphi(p_ 1) \Vert_\infty = 1$. Therefore the norm of $\varphi$ related to $\Vert \cdot \Vert_\infty$ is equal to $1$.
Regarding (2) and denoting $\Vert p \Vert_1$ the norm $\int_0^1 \vert p(t) \vert \ dt$, we have:
$$\Vert \varphi(p) \Vert_1 = \int_0^1 \left\vert \int_0^t p(s) \ ds \right\vert \ dt \le \int_0^1 \left( \int_0^t \vert p(s) \vert \ ds \right) \ dt \le \int_0^1 \left( \int_0^1 \vert p(s) \vert \ ds \right) \ dt = \Vert p \Vert_1.$$ So $\varphi$ is also continuous related to the norm $\Vert p \Vert_1$.
