Elementary Properties of an Orthonormal Family

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space, with zero vector $0_{V}$ and induced norm $\lVert\cdot\rVert$. Let $n$ be a natural number, let $[n]$ be the initial segment determined by $n$, and let $e\in V^{n}$ be an $n$-tuple in $V$ that is orthonormal, with components $e_{k}$. Let $c\in\mathbb{C}^{n}$ be an $n$-tuple with components $c_{k}$ in the field $\mathbb{C}$ of complex numbers, and let $u\in V$.

Sums of vectors are finite sums in $V$ and sums of scalars are finite sums in a field; $|z|$ denotes the modulus of a complex number $z$ and $\overline{z}$ its conjugate; and $x-y$ abbreviates $x+(-y)$, where $-y$ is the additive inverse of $y$ in $V$. Then the following hold.

1. (Coefficients) For every $j\in[n]$,

$$\Bigl\langle e_{j},\sum_{k=1}^{n}c_{k}e_{k}\Bigr\rangle=c_{j}.$$

2. (Norm of a linear combination)

$$\Bigl\lVert\sum_{k=1}^{n}c_{k}e_{k}\Bigr\rVert^{2}=\sum_{k=1}^{n}|c_{k}|^{2},$$

the sum on the right being a finite sum of real numbers.

3. (Linear independence) The tuple $e$ is linearly independent.

4. (Orthogonal decomposition) Put

$$p=\sum_{k=1}^{n}\langle e_{k},u\rangle e_{k},\qquad w=u-p .$$

Then $u=w+p$, and $\langle e_{j},w\rangle=0$ for every $j\in[n]$, and $p$ and $w$ are orthogonal, and

$$\lVert u\rVert^{2}=\sum_{k=1}^{n}\bigl|\langle e_{k},u\rangle\bigr|^{2}+\lVert w\rVert^{2}.$$

5. (Bessel's inequality)

$$\sum_{k=1}^{n}\bigl|\langle e_{k},u\rangle\bigr|^{2}\le\lVert u\rVert^{2},$$

an inequality between real numbers in the order of the ordered field $\mathbb{R}$.