Non Equivalent Norms

This one’s diehard.

Every two norms on a finite dimensional vector space are equivalent: that is to say, each one can be rescaled by a constant to bound the other from both above and below. A consequence of this is that all norms induce the same topology.

In infinite dimension, this is not true anymore. But so far I have only seen examples of pair of norms where one beats the other on the whole space.

Is there a computable example of two norms on a (necessarily infinite dimensional) (topological, locally convex, whatever) vector space such that their difference changes sign over different points?

I’m already aware that if you are pro-choice you can cook up unspeakable ordinals of inequivalent norms using Hamel basis and unbounded operators. But sometimes one needs to value life too.