Uniqueness in the Gram-Schmidt process with positive normalization ($⟨w_j,u_j⟩>0$)

For a linearly independent ordered family in a real inner product space, require each unit vector $w_j$ to lie in the span of the first $j$ inputs and be orthogonal to all earlier inputs. That orthogonal part of the successive span is one-dimensional. Subtracting the earlier orthonormal components gives its nonzero generator $r_j$. Every possible unit choice is therefore $\pm r_j/\Vert r_j\Vert$. Since $⟨r_j,u_j⟩=\Vert r_j{\Vert }^2>0$, the positive normalization fixes the sign uniquely. This distinguishes uniqueness of the normalized ordered construction from the many possible orthonormal bases of the whole space.