= Solution
An \b[<orthonormal basis>] $(q_1,\ldots,q_n)$ is a <basis> with $\langle q_i,q_j\rangle=\delta_{ij}$, where $\delta_{ij}$ is the <Kronecker delta>. Each <vector> has <norm> one and distinct <vectors> are <orthogonal>.
The <Gram-Schmidt process> starts from a <basis> $(e_1,\ldots,e_n)$ and defines
$$
u_1=e_1,\qquad q_1=\frac{u_1}{\|u_1\|},\qquad
u_k=e_k-\sum_{j=1}^{k-1}\langle e_k,q_j\rangle q_j,\qquad
q_k=\frac{u_k}{\|u_k\|}.
$$
The <coefficients> are in this order because the <inner product> is linear in its first argument. For $i<k$, taking the <inner product> with $q_i$ gives $\langle u_k,q_i\rangle=0$. The residual $u_k$ cannot be zero: otherwise $e_k$ would belong to the <linear span> of $e_1,\ldots,e_{k-1}$, contradicting the initial <basis>. Normalization makes $\|q_k\|=1$, while induction preserves
$$
\operatorname{span}(q_1,\ldots,q_k)=\operatorname{span}(e_1,\ldots,e_k).
$$
Thus the process produces an <orthonormal basis> of all of $V$.
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