Given an infinite dimensional normed linear space, how would one show that it is possible to fit an infinite collection of non-overlapping balls of radius $\frac{1}{4}$ in the unit ball?
I guess one can immediately reduce the problem to a normed linear space of countably infinite dimensions. The solution seems clear if the concept of orthogonality exists, but not every normed linear space has an inner product so it's not possible to apply something like Gram-Schmidt to produce an orthogonal basis. Is there any way around this, or there another approach that can be used?