Every countable discrete subgroup of a normed space is isomorphic to the direct sum of the group of integers. I wonder whether it is possible to push this beyond such direct-sum (free abelian) groups in the non-separable case.

Given the above, every discrete subgroup of a normed space is almost free but so is the Baer-Specker group, so it is not enough to look at countable subgroups. I wonder whether the following line of reasoning has a chance of being useful, yet I am not that familiar with absoluteness that much, so likely it will be rubbish.

Shelah proved that non-free Whitehead groups do exist under Martin's Axiom and the negation of CH, but all of them are free under V = L. (These slides nicely explain that.) Being a subset of a normed space seems quite absolute to me. Is it possible then to start with a model where all Whitehead groups are free to conclude that in some model, there are discrete subgroups of normed spaces that are not?

Every countable discrete subgroup of a normed space is isomorphic to the direct sum of the group of integers. I wonder whether it is possible to push this beyond such direct-sum (free abelian) groups in the non-separable case.

Given the above, every countable discrete subgroup of a normed space is almost free but so is the Baer-Specker group, so it is not enough to look at countable subgroups. I wonder whether the following line of reasoning has a chance of being useful, yet I am not that familiar with absoluteness that much, so likely it will be rubbish.

Shelah proved that non-free Whitehead groups do exist under Martin's Axiom and the negation of CH, but all of them are free under V = L. (These slides nicely explain that.) Being a subset of a normed space seems quite absolute to me. Is it possible then to start with a model where all Whitehead groups are free to conclude that in some model, there are discrete subgroups of normed spaces that are not?