Orthogonality, Orthogonal Complement and Orthonormal Families in a Real Inner Product Space

Let $E$ be a real inner product space with inner product $\langle\cdot,\cdot\rangle$ and norm $|\cdot|$, and let $\mathbb{N}$ be the set of natural numbers.

1. (Orthogonal vectors)(i) Two elements $x,y\in E$ are orthogonal, written $x\perp y$, if $\langle x,y\rangle=0$.

2. (Orthogonal complement)(ii) For a subset $M\subseteq E$, the orthogonal complement of $M$ is the subset

$$M^{\perp}=\{\,x\in E\ :\ \langle x,m\rangle=0\ \text{for every }m\in M\,\}$$

of $E$.

3. (Orthonormal families)(iii) For $n\in\mathbb{N}$, an $n$-tuple $e\in E^{n}$ with components $e_{1},\dots,e_{n}$ is orthonormal if $\langle e_{i},e_{j}\rangle=0$ for all $i,j$ in the initial segment $[n]$ with $i\ne j$ and $|e_{i}|=1$ for every $i\in[n]$. A sequence $(e_{k})_{k\in\mathbb{N}}$ in $E$ is orthonormal if $\langle e_{i},e_{j}\rangle=0$ for all $i,j\in\mathbb{N}$ with $i\ne j$ and $|e_{i}|=1$ for every $i\in\mathbb{N}$.