Determine if these two norms are equivalent

Question

Let we have the space $C[a,b]$ (the space of all functions that are continuous on closed interval $[a,b]$). And we have two norms on this space:

$$\|X\|_1= \max_{t\in [a,b]} | x(t) |$$ $$\|X\|_2 = \left( \int_a^b | x(t) |^2 dt \right)^{1/2}$$

Are these norms on this space equivalent or not equivalent and why ?

Answer 1

Let me do it in $[0,1]$. Any interval is the same.

Notice that $f_n(x)=0$ for $1\geq x>1/n$ and $f_n(x)=-nx+1$ for $x\in[0,1/n]$ is Cauchy in $||\cdot||_2$ but not in $||\cdot||_1$

We can check it by computing a bit. Let $0$ be the zero function when appropriate.

$$||f_n-0||_1=||f_n||_1=max |f| = 1$$

while

$$||f_n-0||_2=||f_n||_2=\left(\int_{0}^{1/n}\left|nx-1\right|^2\right)^{1/2}=\left(\frac{1}{3n^2}\right)^{1/2}\to 0$$