you cannot do that: the euclidean norm is not defined for an infinite-dimensional space.

Why not? It is the square root of the sum of (dxi)^2, where i goes through all dimensions. Sometimes it is a finite value. Otherwise the distance is infinite.

The points T0(0,0,0,0....) and T1(0,1/sqrt(2),1/sqrt(4),1/sqrt(8)...) are 1 apart.

I'll try to steelman Florian_Dietz.

I don't know much anything about relativity, but waves on a grid in computational fluid dynamics (CFD for short) typically don't have the problem you describe. I do vaguely recall some strange methods that do in a Lagrangian CFD class I took, but they are definitely non-standard and I think were used merely as simple illustrations of a class of methods.

Plus, some CFD methods like the numerical method of characteristics discretize in different coordinates that follow the waves. This can resolve waves really well, but it's confusing to set up in higher dimensions.

CFD methods are just particularly well developed numerical methods for physics. From what I understand analogous methods are used for computational physics in other domains (even relativity).

Not even for wavelengths not much longer than the grid spacing?