This is a problem from a past qualifying exam for applied analysis.
Consider the $1$, $2$ and $\infty$-norms on the usual Banach Space of continuous functions on $[0,1]$.
Under which of these norms is $C[0,1]$ complete?
Determine which norms are stronger than which others. Are any of them equivalent?
Clearly, $C[0,1]$ is complete under the $\infty$-norm--I don't have trouble proving this. I'm relatively certain that it is not complete under the $L^1$ and $L^2$ norms, but I'm not exactly sure how to show this.
For 2, I'm a bit confused about what it means to show one norm stronger than another. I have a clear definition for two norms being equivalent, but does $||\cdot||_1$ is stronger than $||\cdot||_2$ just mean there is some constant $c>0$ such that $||x||_2 \leq c ||x||_1$ for all $x$ in the appropriate space?