$P_3$ is unbounded on ($C[0,1], \lVert \cdot \rVert_2$)

Question

Is it true that $P_3$, the set of all polynomials of degree 3, is unbounded on ($C[0,1], \lVert \cdot \rVert_2$), the set of all continuous function on $[0,1]$? I think it should be true because if $f(x)=a+bx+cx^2+dx^3$, then $a,b,c,d\in \mathbb{R}$ can be any real numbers, and so the 2-norm of $f$ will not be bounded. Please correct me if I'm wrong, maybe I'm missing something.

Answer 1

You're right. To show it just notice that if $p(x)=x^3$ and $a\neq0$ then $ap\in P_3$ and $$ ||ap||_2^2=a^2\int_0^1x^6dx=\frac{a^2}{7}. $$ So by taking $a$ large enough you can make $||ap||_2$ as large as you want; i.e. $P_3$ is not bounded with $||\cdot||_2$.