Orthonormal Expansions in a Real Hilbert Space

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

1. (Bessel's inequality)(i) The series $\sum_{k=1}^{\infty}\langle x,e_{k}\rangle^{2}$ converges, and

$$\sum_{k=1}^{\infty}\langle x,e_{k}\rangle^{2}\le|x|^{2}.$$

2. (Riesz-Fischer criterion)(ii) The series $\sum_{k=1}^{\infty}c_{k}e_{k}$ converges in $H$ if and only if the series $\sum_{k=1}^{\infty}c_{k}^{2}$ converges. In that case, writing $z=\sum_{k=1}^{\infty}c_{k}e_{k}$,

$$|z|^{2}=\sum_{k=1}^{\infty}c_{k}^{2}, \qquad \langle z,e_{j}\rangle=c_{j}\quad\text{for every }j\in\mathbb{N}.$$

3. (Expansion)(iii) Suppose $(e_{k})_{k\in\mathbb{N}}$ is an orthonormal basis of $H$. Then the series $\sum_{k=1}^{\infty}\langle x,e_{k}\rangle e_{k}$ converges in $H$ with sum $x$.

4. (Parseval's identity)(iv) Suppose $(e_{k})_{k\in\mathbb{N}}$ is an orthonormal basis of $H$. Then

$$|x|^{2}=\sum_{k=1}^{\infty}\langle x,e_{k}\rangle^{2},$$

and the series $\sum_{k=1}^{\infty}\langle x,e_{k}\rangle\langle y,e_{k}\rangle$ converges with sum $\langle x,y\rangle$.

5. (Characterisation)(v) The following four conditions on the orthonormal sequence $(e_{k})_{k\in\mathbb{N}}$ are equivalent:

(a) $(e_{k})_{k\in\mathbb{N}}$ is an orthonormal basis of $H$;

(b) every $x\in H$ is the sum of the series $\sum_{k=1}^{\infty}\langle x,e_{k}\rangle e_{k}$;

(c) every $x\in H$ satisfies $|x|^{2}=\sum_{k=1}^{\infty}\langle x,e_{k}\rangle^{2}$;

(d) the union of the subspaces $\operatorname{span}(e^{(n)})$ over $n\in\mathbb{N}$ is dense in $H$.