Let $(U, \|-\|_u), (V, \|-\|_v)$ be any to normed linear spaces. A norm on $U\times V$ is called a product norm if when restricted to $U$ and $V$ it agrees with $\|-\|_u$ and $\|-\|_v$ respectively. We investigate existence inequivalent product norms. We hope that this will help to understand the space equivalence classes of all norms on $\ell_1(N).$
