Norm of an operator on space of real polynomials Let $L:\mathbb{R}[X]\rightarrow\mathbb{R}[X]$ be an operator given by the following formula
$L\left(\sum\limits_n a_nX^n\right)=\sum\limits_n a_{2n}X^{2n}$.
We assume that on $\mathbb{R}[X]$ we have a norm $\|p\|=\int\limits_{-1}^{1}|p(t)|dt$.
Is this operator bounded ? If yes what is its norm ?
I tried to do it in a different ways, but all my ideas failed. 
 A: [Answer under repair; see comments below]

Take $p = p_{n}(x) = x^{2n} - x^{2n+1}$, so that $L(p) = x^{2n}$.  We note that
$$
\|p\| = \int_{-1}^1 |p(t)|\,dt = 2\int_0^1 [t^{2n} - t^{2n+1}]\,dt = 2 \left[ \frac {1}{2n} - \frac{1}{2n+1}\right] = \frac{1}{n(2n+1)}
$$
On the other hand,
$$
\|L(p)\| = \int_{-1}^1 t^{2n}\,dt = \frac{2}{2n+1}
$$
From there, it remains to be shown that $\left(\frac{p_n}{\|p_n\|}\right)_{n=1}^\infty$ is a bounded sequence for which the image under $L$ is unbounded.
A: for any two polynomials $P(x),Q(x)$ :
$$L\left(P(x^2)+ x Q(x^2)\right) = P(x^2)$$
now think to :
$$\min_{Q} \quad||P(x^2) + x Q(x^2)|| = \int_{-1}^1 \left|P(x^2)+x Q(x^2)\right| dx$$
because $P(x^2)$ is an even function while $x Q(x^2)$ is an odd function, the minimum is attained for $Q(x^2) = 0$.
thus :
$$||L(P(x^2)+x Q(x^2))||=||P(x^2)||\le ||P(x^2)+xQ(x^2)||$$
so the operator is bounded (which as shown in the discussion wouldn't be the case if we were considering the $\displaystyle\int_0^1 \,|.|\, dx$ norm instead !) and clearly its norm is $||L|| = 1$
note : that a projection operator on a Banach space is bounded or not seems to be related to the Schauder basis, so I would be glad if someone could explain in term of Shauder basis why $L$ is bounded for the $\int_{-1}^1 |.|dx$ norm but not for the $\int_0^1 |.|dx$
