= Uniqueness in the Gram-Schmidt process with positive normalization
{title2=$\langle w_j,u_j\rangle>0$}
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 $\pm r_j/\|r_j\|$. Since $\langle r_j,u_j\rangle=\|r_j\|^2>0$, 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.
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