The Subspaces Spanned by an Orthonormal Sequence and Exhausting Sequences

In the setting of Real Hilbert Spaces: Standing Notation and Background, let $H$ be a real Hilbert space with inner product $\langle\cdot,\cdot\rangle$ and norm $|\cdot|$, and let $(e_{k})_{k\in\mathbb{N}}$ be an orthonormal sequence in $H$. For $n\in\mathbb{N}$ let $e^{(n)}\in H^{n}$ be the $n$-tuple whose components are $e_{1},\dots,e_{n}$ and let $H_{n}=\operatorname{span}(e^{(n)})$ be its span. Then the following hold.

1. (The spanned subspaces)(i) For every $n\in\mathbb{N}$, the set $H_{n}$ is a closed linear subspace of $H$ and is finite-dimensional as a vector space over $\mathbb{R}$, and $H_{n}\subseteq H_{n+1}$. The orthogonal projection $P_{H_{n}}$ onto $H_{n}$ satisfies

$$P_{H_{n}}x=\sum_{k=1}^{n}\langle x,e_{k}\rangle e_{k}\qquad\text{for every }x\in H .$$

2. (Exhausting sequences)(ii) The sequence $(H_{n})_{n\in\mathbb{N}}$ is an exhausting sequence for $H$ if and only if $(e_{k})_{k\in\mathbb{N}}$ is an orthonormal basis of $H$.