I have a regular curve, , in ℝ^{N}(parameterised by its arc length, ). I am trying to define the moving (Frenet) frame of orthonormal vectors . However, looking in different books, I find subtly different definitions (both based on Gram-Schmidt orthogonalisation). I believe the two methods (described in full below) are equivalent, essentially because is a linear combination of the derivatives . However, I would like to be absolutely sure. To sum up, my question is:

Do the following two approaches yield the same result?

... suggested in, for example, [1, p. 13] (link) and [2] (link).

... suggested in, for example, [3, p. 159].

In other words, is the subspace spanned by the same as the subspace spanned by ?

References:

[1] W. Kühnel, "Differential Geometry: Curves - Surfaces - Manifolds".

[2] Wikipedia, "Frenet–Serret formulas".

[3] H. W. Guggenheimer, "Differential Geometry", McGraw Hill (or Dover Edition), 1963 (1977).