Operator on polynomials, antiderivative We are given a linear map:
$$\mathbb{R} [X] \ni p \rightarrow q \in \mathbb{R} [X],  \ \ q'=p, \ \ q(0)=0 $$
and two norms on $ \mathbb{R} [X]$ : $||p||_{\infty} = \sup _{t \in [0,1]} |p(t)|$, $||p||_1  = \int_0^1 |p(t)|dt$ 
I want to check whether the map is continuous in these norms.
When it comes to the first norm, I get:
$$||q||_{\infty} = \sup _{t \in [0,1]} |q(t)|$$ By Lagrange mean value theorem we have that there exists $a \in [0,1]$ such that $q'(a) = \frac{q(t)-q(0)}{t}$, so $q(t) \le \sup_{[0,1]} |q'(t)|$
And $||p||_{\infty} = \sup _{t \in [0,1]} |p(t)| = \sup _{t \in [0,1]} |q'(t)| \ge ||q||$
Is that correct so far?
What can I do about the other norm?

Your work is correct, but in my opinion this makes things more clear:
$$
q(x)=\int_0^xp(t)\,dt.
$$
Then, for the $\|\,\|_\infty$ norm
$$
|q(x)|\le\int_0^x|p(t)|\,dt\le x\,\|p\|_\infty\le\|p\|_\infty.
$$
For the $\|\,\|_1$ norm
$$
\int_0^1|q(x)|\,dx=\int_0^1\Bigl|\int_0^xp(t)\,dt\Bigr|\,dx\le\int_0^1\int_0^1|p(t)|\,dt\,dx=\|p\|_1.
$$
