I was explaining to my students that if there is an inequality between two norms, then there is an inclusion between their spaces of convergent sequences, *with matching limits*. I then proceeded to show examples of such inequalities on the normed spaces they knew, and counterexamples of sequences which converge for a norm and not for another, stating the equivalence of norms in finite dimension, etc.

It is then that I wondered about the following : does there exist a vector space, two norms on that vector space and a single sequence which converges for both norms, but with **different** limits?

The first remark is that such a counter-example cannot exist in finite dimension ; and one first has to find "really inequivalent norms", which do exist : consider the space of polynomials in one variable, and define norms on it by summing the absolute values of the coefficients :

- first with a weight $1$ for every coefficient ;
- second with $2^n$ or $2^{-n}$ depending on the parity of the degree $n$.

It's now easy to find a sequence going to zero for the first and not for the second, and a sequence going to zero for the second and not for the first - so there can't be an inequality between those.

Notice this is all over the real or complex numbers, though the question could be amusing in a more general setting.