Agnus Dei, qui tollis peccata mundi.
\section{Ortonormalisasi Gram-Schmidt}
Andaikan ada sebuah ruang vektor $\mathcal{H}$ di atas lapangan $\mathbb{C}$. Andaikan ada seperangkat vektor $B_n := \{ \psi_1, \cdots, \psi_n \} \subset \mathcal{H}$ yang bebas linier. Ortonormalisasi Gram-Schmidt dari $B_n$ menghasilkan seperangkat vektor $O_n := \{ \varphi_1, \cdots, \varphi_n \} \subset \mathcal{H}$ yang ortonormal. Andaikan didefinisikan sebuah produk skalar $\left<\alpha|\beta\right> \in \mathbb{C}$ untuk semua $\alpha, \beta \in \mathcal{H}$. Andaikan pula didefinisikan sebuah norma $\|\alpha\| := \sqrt{\left<\alpha|\alpha\right>} \in \mathbb{C}$ untuk semua $\alpha \in \mathcal{H}$.
Mula-mula, $\varphi_1 := \psi_1/\|\psi_1\|$.
Selanjutnya,
\[ \varphi_j := \frac{1}{N_j}\left(\psi_j - \sum_{k = 1}^{j - 1} \left<\varphi_k|\psi_j\right>\varphi_k\right) \]
di mana
\[ N_j := \left\|\psi_j - \sum_{k = 1}^{j - 1} \left<\varphi_k|\psi_j\right>\varphi_k\right\| \]
untuk setiap $j \in \{ 2, \cdots, n \}$.
Ternyata, $\left<\varphi_j|\varphi_l\right> = \delta_{jl}$ untuk setiap $j, l \in \{ 1, \cdots, n \}$.
Pembuktian hal ini secara umum sangat rumit, mengingat terdapat ungkapan rekursif. Namun, kita akan membuktikan kasus khusus untuk $n = 3$.
Tentu jelas bahwa $\left<\varphi_j|\varphi_j\right> = \|\varphi_j\|^2 = N_j^2/N_j^2 = 1$ untuk semua $j \in \{ 1, 2, 3 \}$.
Sekarang, kita tinggal membuktikan bahwa $\left<\varphi_1|\varphi_2\right> = \left<\varphi_1|\varphi_3\right> = \left<\varphi_2|\varphi_3\right> = 0$.
Mula-mula kita tulis secara eksplisit bahwa
\[ \varphi_2 = \frac{1}{N_2}\left(\psi_2 - \left<\varphi_1|\psi_2\right>\varphi_1\right) \]
dan
\[ \varphi_3 = \frac{1}{N_3}\left(\psi_3 - \left<\varphi_1|\psi_3\right>\varphi_1 - \left<\varphi_2|\psi_3\right>\varphi_2\right). \]
\[ \left<\varphi_1|\varphi_2\right> = \frac{1}{N_2}\left(\left<\varphi_1|\psi_2\right> - \left<\varphi_1|\psi_2\right>\left<\varphi_1|\varphi_1\right>\right) = 0. \]
\[ \left<\varphi_1|\varphi_3\right> = \frac{1}{N_3}\left(\left<\varphi_1|\psi_3\right> - \left<\varphi_1|\psi_3\right>\left<\varphi_1|\varphi_1\right> - \left<\varphi_2|\psi_3\right>\left<\varphi_1|\varphi_2\right>\right) = 0. \]
\[ \left<\varphi_2|\varphi_3\right> = \frac{1}{N_2N_3}\left(\left<\psi_2|\psi_3\right> - \left<\varphi_1|\psi_3\right>\left<\psi_2|\varphi_1\right>\right. \]
\[ - \left<\varphi_2|\psi_3\right>\left<\psi_2|\varphi_2\right> - \left<\psi_2|\varphi_1\right>\left<\varphi_1|\psi_3\right> \]
\[ \left.+ \left<\psi_2|\varphi_1\right>\left<\varphi_1|\psi_3\right> \right) \]
\[ = \frac{1}{N_2N_3}\left(\left<\psi_2|\psi_3\right> - \left<\varphi_1|\psi_3\right>\left<\psi_2|\varphi_1\right>\right. \]
\[ - \frac{1}{N_2^2}\left(\left<\psi_2|\psi_3\right> - \left<\psi_2|\varphi_1\right>\left<\varphi_1|\psi_3\right>\right) \]
\[ \left.\left(\|\psi_2\|^2 - \left<\varphi_1|\psi_2\right>\left<\psi_2|\varphi_1\right>\right)\right). \]
Karena $N_2^2 = \|\psi_2\|^2 - |\left<\varphi_1|\psi_2\right>|^2$, maka
\[ \left<\varphi_2|\varphi_3\right> = \frac{1}{N_2N_3}\left(\left<\psi_2|\psi_3\right> - \left<\psi_2|\psi_3\right> - \left<\psi_2|\varphi_1\right>\left<\varphi_1|\psi_3\right> \right. \]
\[ \left.+ \left<\psi_2|\varphi_1\right>\left<\varphi_1|\psi_3\right>\right) = 0. \]
Om santi santi om.
