Different norms on a product space $X \times Y$ It is well known how to define standard product topology on a product space $\prod_{i \in I} X_i$.
Assume now that $(X,\lVert \, \cdot \, \rVert_{X})$ and $(Y,\lVert \, \cdot \, \rVert_{Y})$ are normed spaces and that the space $X \times Y$ is also equipped with a norm $\lVert \, \cdot \, \rVert_{X \times Y}$.
Is it true that all norms on $X \times Y$ are equivalent?
It is quite easy to prove this if $\lVert \, \cdot \, \rVert_{X \times Y}$ is one of the p-norms, i.e. $\lVert (x,y) \rVert_p = (\lVert x \rVert_X^p + \lVert y \rVert_Y^p)^{1/p}$. All such norms are equivalent. We only need to know that all norms on a finite dimensional space are equivalent (in this case we use it for $\mathbb{R}^2$).
How it is general case?

It is not true in general. When the vector space if finitely dimensional all norms are always equivalent, but when the space has infinite dimension (for example the space of continuous functions on the reals) then norms don't have to be equivalent.
