Operator on polynomials, antiderivativeWe are given a linear map: R [ X ] ∋ p...

Operator on polynomials, antiderivative
We are given a linear map:
$\mathbb{R}[X]\ni p\to q\in \mathbb{R}[X],q^{\prime }=p,q(0)=0$ and two norms on $\mathbb{R}[X]$ : $||p|{|}_{\mathrm{\infty }}=\underset{t\in [0,1]}{sup}|p(t)|$, $||p|{|}_1={\int }_0^1|p(t)|dt$.
I want to check whether the map is continuous in these norms.
When it comes to the first norm, I get:
$||q|{|}_{\mathrm{\infty }}=\underset{t\in [0,1]}{sup}|q(t)|$
By Lagrange mean value theorem we have that there exists $a\in [0,1]$ such that $q^{\prime }(a)=\frac{q(t)-q(0)}{t}$, so $q(t)\le \underset{[0,1]}{sup}|q^{\prime }(t)|$.
And $||p|{|}_{\mathrm{\infty }}=\underset{t\in [0,1]}{sup}|p(t)|=\underset{t\in [0,1]}{sup}|q^{\prime }(t)|\ge ||q||$
Is that correct so far?