Suppose $(X_{i}, \Vert \cdot \Vert_i)_{i\in I}$ are all normed spaces over the same field $\Phi= \mathbb{R}, \mathbb{C}$ and suppose $X= \prod_{i \in I} X_i$ is the product space. I want to show that the product topology on $X$ is normable iff $I$ is finite. My work in the '$\Leftarrow$'-direction, is to show that the bases for the two topologies are equal and so therefore generate the same topology.
In the other direction however, I seem to get stuck, since I think the right way to go about it is to suppose that $I$ is infinite, but that the product topology on $X$ is normable and try to reach a contradiction, but I can't figure out where the contradiction is supposed to come from?