Question

Is there some absolute orthonormed system in space?

Answer 1

If you have a vector space VV augmented with an inner product [1], then you can construct sets of vectors S:={vi}S:={vi}, which are mutually orthogonal [2], i.e. have an inner product of zero:

∀vi,vj∈S,i≠j:⟨vi,vj⟩=0

Answer 2

Explanation:

can be proved that the cardinality of a complete orthonormal system is an invariant of the space (its dimension) and that spaces of the same dimension are linearly isometric. For a proof of this fact, see, for example, Akhiezer and Glazman