Define what it means for two norms on a real vector space $V$ to be Lipschitz equivalent. Show that if two norms on $V$ are Lipschitz equivalent and $F \subset V$, then $F$ is closed in one norm if and only if $F$ is closed in the other norm.

Show that if $V$ is finite-dimensional, then any two norms on $V$ are Lipschitz equivalent.

Show that $\|f\|_{1}=\int_{0}^{1}|f(x)| d x$ is a norm on the space $C[0,1]$ of continuous realvalued functions on $[0,1]$. Is the set $S=\{f \in C[0,1]: f(1 / 2)=0\}$ closed in the norm $\|\cdot\| 1$ ?

Determine whether or not the norm $\|\cdot\|_{1}$ is Lipschitz equivalent to the uniform $\operatorname{norm}\|\cdot\|_{\infty}$ on $C[0,1]$.

[You may assume the Bolzano-Weierstrass theorem for sequences in $\mathbb{R}^{n}$.]