Let $\mathbb{F}\in\{\mathbb{R},\mathbb{C}\}$, and let $V$ be an [inner product space](/page/Inner%20Product%20Space) over $\mathbb{F}$ with [inner product](/page/Inner%20Product) $(\cdot,\cdot)_V$ linear in the first argument and norm $|x|=(x,x)_V^{1/2}$. Let $m\in\mathbb{N}$, and let $(v_1,\ldots,v_m)$ be a linearly independent list in $V$.
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Define the classical Gram-Schmidt residuals and normalized vectors recursively by
for $1\le k\le m$, whenever the residual is nonzero. Then, in exact arithmetic, every residual $u_{C,k}$ and $u_{M,k}$ is nonzero, and $e_{C,k}=e_{M,k}$ for every $1\le k\le m$.