Proof of the existence of an orthonormal basis by the Gram–Schmidt process

Proof

We first prove the extension statement. The existence then follows by extending the empty family.

1. Let $V$ be an inner product space over $\R$ or $\C$ of finite dimension $n$, with inner product $\ip{\cdot}{\cdot}$ and induced norm $\norm{\cdot}$.
2. Let $(e_i)_{i \in I}$ be an orthonormal family in $V$. Its vectors are distinct, since $\ip{e_i}{e_j} = 0 \ne 1 = \ip{e_i}{e_i}$ for $i \ne j$.
3. The family is linearly independent. So its set of vectors extends to a basis $B$ of $V$.
4. All bases of $V$ have the same cardinality, so $B$ has $n$ elements. Hence the family is finite. Write it $e_1, \dots, e_m$ with $m \le n$.
5. List $B$ as $v_1, \dots, v_n$ with $v_i = e_i$ for $i \le m$. Let $u_1, \dots, u_n$ be the vectors of the Gram–Schmidt process applied to $v_1, \dots, v_n$.
6. We show $u_j = e_j$ for $j \le m$, by induction on $j$. First, $u_1 = v_1 = e_1$.
7. Let $2 \le j \le m$ and assume $u_i = e_i$ for $i < j$. Then $u_j = e_j - \sum_{i<j} \frac{\ip{e_j}{e_i}}{\ip{e_i}{e_i}} e_i = e_j$, because $\ip{e_j}{e_i} = 0$ for $i \ne j$.
8. By the properties of the Gram–Schmidt vectors, $u_1, \dots, u_n$ are nonzero and pairwise orthogonal. Also $\operatorname{span}(u_1, \dots, u_n) = \operatorname{span}(v_1, \dots, v_n) = V$.
9. Set $f_j = u_j / \norm{u_j}$, which is defined since $u_j \ne 0$. Then $\ip{f_j}{f_j} = \ip{u_j}{u_j} / \norm{u_j}^2 = 1$.
10. For $i \ne j$, $\ip{f_i}{f_j} = \ip{u_i}{u_j} / (\norm{u_i} \norm{u_j}) = 0$, since $\norm{u_j}$ is real. So $f_1, \dots, f_n$ is an orthonormal family.
11. For $j \le m$, $f_j = e_j / \norm{e_j} = e_j$, since $\norm{e_j} = 1$.
12. The family $f_1, \dots, f_n$ is linearly independent by step 10 and the same result as in step 3. It has $n = \dim V$ vectors, so it is a basis of $V$.
13. So $f_1, \dots, f_n$ is an orthonormal basis of $V$ that extends $e_1, \dots, e_m$. This proves the second claim.
14. The empty family is orthonormal, because the condition on pairs is vacuous. Applying step 13 to it gives an orthonormal basis of $V$. This proves the first claim.