I have a regular curve, $\displaystyle \underline{a}(s)$, in ℝ^{N}(parameterised by its arc length, $\displaystyle s$). I am trying to define the moving (Frenet) frame of orthonormal vectors $\displaystyle \left\{\underline{u}_1(s),\underline{u}_2(s),\dots , \underline{u}_N(s) \right\}$. 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 $\displaystyle \underline{u}_{k-1}^{\prime}(s)$ is a linear combination of the derivatives $\displaystyle \underline{a}^{\prime}(s), \underline{a}^{ \prime \prime}(s), \dots, \underline{a}^{(k)}(s)$. However, I would like to be absolutely sure. To sum up, my question is:$\displaystyle \underline{u}_k(s)=\frac{\underline{a}^{(k)}(s) - \sum\limits_{m=1}^{k-1}\left(\underline{u}_m^T(s)\underline{a}^{(k)}(s) \right) \underline{u}_m(s)}{\Vert numerator \Vert}$

Do the following two approaches yield the same result?

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

$\displaystyle \underline{u}_k(s)=\frac{\underline{u}_{k-1}^{\prime}(s) - \sum\limits_{m=1}^{k-1}\left(\underline{u}_m^T(s)\underline{u}_{k-1}^{\prime}(s) \right) \underline{u}_m(s)}{\Vert numerator \Vert}$

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

In other words, is the subspace spanned by $\displaystyle \left\{\underline{a}^{\prime}, \underline{a}^{ \prime \prime}, \dots, \underline{a}^{(k)}\right\}$ the same as the subspace spanned by $\displaystyle \left\{\underline{u}_1, \underline{u}_2, \dots, \underline{u}_{k-1}, \underline{u}_{k-1}^{\prime} \right\}$?

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).