Is $L^{p}$ space with alternate norm Banach?
$E$ is a measurable set of finite measure and $1 \le p_{1} < p_{2} < \infty$. Consider the linear space $L^{p_{2}}(E)$ normed by $||.||_{p_{1}}$. Is this space Banach?
Assuming $E$ does not consist of a finite number of atoms (in which case you have a finite-dimensional vector space, where all norms are equivalent), there is a sequence of measurable subsets $A_j$ of $E$ with $m(A_j) > 0$ and $m(A_j) \to 0$. Consider the indicator functions of such sets, and use the Open Mapping Theorem.