Showing norm of a continuous space is not equivalent

Question

Question: Show that on $C\left ( \left [ a,b \right ],\mathbb{R} \right ) $the uniform norm $\left | \cdot \right |_{0}$ and the $L^{2} Norm \left \| \cdot \right \|_{p=2}$ are not equivalent norm.

Suppose there exists constant c>0 such that $\left | f \right |_{0} \leq c\left \| f \right \|_{p=2}$. I seek to find a continuous function which contradicts this equality.

Any help is appreciated.

Thanks in advance.

Answer 1

Hint: think of functions whose graphs have a tall and narrow spike.

Answer 2

Try $f_n(x) = x^n$ on $[0,1]$.