Projection onto the Span of an Orthonormal Tuple, and Coordinates on a Finite-Dimensional Subspace

Let $\mathbb{R}$ be the ordered field of real numbers, with the notation of that item, and let $\mathbb{N}$ be the set of natural numbers. Let $E$ be a real inner product space with inner product $\langle\cdot,\cdot\rangle$, norm $|\cdot|$, distance $d$, let $m\in\mathbb{N}$, let $e\in E^{m}$ be an orthonormal $m$-tuple with components $e_{i}$, and let $M=\operatorname{span}(e)$ be its span, a linear subspace of $E$ by claim 1 of The Span of a Finite Family is the Smallest Subspace Containing It and hence a vector space over $\mathbb{R}$ under the restricted operations by claim 1 of A Linear Subspace is a Vector Space and Inherits an Inner Product. Let $M^{\perp}$ be the orthogonal complement of $M$. Sums of vectors are finite sums in $E$ and sums of real numbers are finite sums in $\mathbb{R}$. Define $P:E\to E$ by

the finite sum of the $m$-tuple with components $\langle x,e_{i}\rangle e_{i}$. Let $\mathbb{R}^{m}$ be Euclidean space, a vector space over $\mathbb{R}$ by Euclidean Space $\mathbb{R}^n$ is a Real Vector Space, with the dot product $\xi\cdot\zeta$ and the Euclidean norm $\lVert\xi\rVert$; a point $\xi\in\mathbb{R}^{m}$ is written $\xi=(\xi_{1},\dots,\xi_{m})$ with real coordinates $\xi_{i}$. Then the following hold.

1. (Projection)¶ $P$ is a linear map, $Px\in M$ for every $x\in E$, $Px=x$ for every $x\in M$, and $x-Px\in M^{\perp}$ for every $x\in E$.

2. (Bessel and Pythagoras)¶ For every $x\in E$, $|Px|^{2}=\sum_{i=1}^{m}\langle x,e_{i}\rangle^{2}$ and $|x|^{2}=|Px|^{2}+|x-Px|^{2}$; consequently $\sum_{i=1}^{m}\langle x,e_{i}\rangle^{2}\le|x|^{2}$, $|Px|\le|x|$ and $|x-Px|\le|x|$.

3. (Nonexpansiveness)¶ For all $x,y\in E$, $|Px-Py|\le|x-y|$; in particular $P$ is Lipschitz with constant $1$ from $(E,d)$ to $(E,d)$.

4. (Closedness)¶ $M$ is closed in $E$.

5. (Nearest point)¶ For every $x\in E$ and every $y\in M$, $|x-Px|\le|x-y|$. If $E$ is a real Hilbert space, then $M$ is a closed linear subspace of $E$ and $P=P_{M}$, the orthogonal projection onto $M$ characterised in Orthogonal Projection onto a Closed Linear Subspace of a Real Hilbert Space §characterisation.

6. (Coordinates)¶ The map $\kappa:M\to\mathbb{R}^{m}$ given by $\kappa(x)=(\langle x,e_{1}\rangle,\dots,\langle x,e_{m}\rangle)$ is a linear bijection, with inverse $\xi\mapsto\sum_{i=1}^{m}\xi_{i}e_{i}$, and for all $x,y\in M$,