Proof of Orthonormal Expansions in a Real Hilbert Space

Throughout, $s_{n}=\sum_{k=1}^{n}c_{k}e_{k}$ denotes the $n$-th partial sum of $(c_{k}e_{k})_{k\in\mathbb{N}}$ in $H$ and $t_{n}=\sum_{k=1}^{n}c_{k}^{2}$ that of $(c_{k}^{2})_{k\in\mathbb{N}}$ in $\mathbb{R}$; $S$ is the successor map and $n+1=S(n)$. For each $n$, the tuple $e^{(n)}$ is orthonormal, since its components are components of the orthonormal sequence $(e_{k})$ and distinct indices in $[n]$ are distinct in $\mathbb{N}$.

Squares are nonnegative. For $s\in\mathbb{R}$, claim 1 of Properties of the Absolute Value in an Ordered Field gives $|s|\in\{s,-s\}$ and $0\le|s|$; in either case $|s|\,|s|=s\,s=s^{2}$, and $0\le|s|\,|s|$ by claim 5 of Elementary Arithmetic in an Ordered Field. Hence $0\le s^{2}$.

Claim 1. Let $P^{(n)}$ denote the projection onto $\operatorname{span}(e^{(n)})$ of Projection onto the Span of an Orthonormal Tuple, and Coordinates on a Finite-Dimensional Subspace. By claim 2 of that lemma,

$$\sum_{k=1}^{n}\langle x,e_{k}\rangle^{2}=|P^{(n)}x|^{2}\le|x|^{2}\qquad\text{for every }n\in\mathbb{N}.$$

The terms $\langle x,e_{k}\rangle^{2}$ are nonnegative, so the set of partial sums of $\sum_{k=1}^{\infty}\langle x,e_{k}\rangle^{2}$ is bounded above by $|x|^{2}$, and claim 1 of Series of Nonnegative Real Numbers, Comparison, and the Geometric Series shows that the series converges with sum the supremum of that set, which is at most $|x|^{2}$.

Claim 2. We first show, by induction on $n$, that

$$|s_{n}-s_{m}|^{2}=t_{n}-t_{m}\qquad\text{whenever }m\le n.$$

If $n=1$ then $m=1$ by claims 4 and 2 of Properties of the Order on the Natural Numbers, and both sides vanish. Assume the identity for $n$ and let $m\le n+1$. If $m=n+1$ both sides vanish. Otherwise $m\le n$ by claim 5 of Properties of the Order on the Natural Numbers. By claim 2 of Inner Products Against Finite Sums, and Orthonormal Families, in a Real Inner Product Space and orthonormality, $\langle s_{n},e_{n+1}\rangle=\sum_{k=1}^{n}c_{k}\langle e_{k},e_{n+1}\rangle=0$, since $k\le n<n+1$ gives $k\ne n+1$; likewise $\langle s_{m},e_{n+1}\rangle=0$, so $\langle s_{n}-s_{m},e_{n+1}\rangle=0$ by Elementary Identities in a Real Inner Product Space (i). Hence, by Elementary Identities in a Real Inner Product Space (v) and $|e_{n+1}|=1$,

$$|s_{n+1}-s_{m}|^{2}=|(s_{n}-s_{m})+c_{n+1}e_{n+1}|^{2}=|s_{n}-s_{m}|^{2}+c_{n+1}^{2}=(t_{n}-t_{m})+c_{n+1}^{2}=t_{n+1}-t_{m}.$$

Suppose $\sum_{k=1}^{\infty}c_{k}^{2}$ converges and let $\varepsilon$ be positive. By claim 3 of Elementary Properties of Series of Real Numbers there is $N$ with $|t_{n}-t_{m}|<\varepsilon^{2}$ for $N\le m$ and $N\le n$. For such $m,n$, the order being total, we may assume $m\le n$ after possibly exchanging them, since $|s_{n}-s_{m}|=|s_{m}-s_{n}|$; then $|s_{n}-s_{m}|^{2}=t_{n}-t_{m}\le|t_{n}-t_{m}|<\varepsilon^{2}$ by claim 3 of Properties of the Absolute Value in an Ordered Field, whence $|s_{n}-s_{m}|<\varepsilon$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field and trichotomy. By claim 4 of Elementary Properties of Series in a Real Inner Product Space the series $\sum_{k=1}^{\infty}c_{k}e_{k}$ converges.

Conversely, suppose $\sum_{k=1}^{\infty}c_{k}e_{k}$ converges and let $\varepsilon$ be positive. Let $\eta$ be the nonnegative real with $\eta^{2}=\varepsilon$ given by Existence and Uniqueness of the Nonnegative Square Root; then $\eta$ is positive. By claim 4 of Elementary Properties of Series in a Real Inner Product Space there is $N$ with $|s_{n}-s_{m}|<\eta$ for $N\le m$ and $N\le n$, and then $|t_{n}-t_{m}|=|s_{n}-s_{m}|^{2}<\varepsilon$ for such $m,n$ with $m\le n$, using claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field; the case $n\le m$ follows by exchanging $m$ and $n$. By claim 3 of Elementary Properties of Series of Real Numbers the series $\sum_{k=1}^{\infty}c_{k}^{2}$ converges.

Assume now both converge and write $z=\sum_{k=1}^{\infty}c_{k}e_{k}$. By claim 4 of Inner Products Against Finite Sums, and Orthonormal Families, in a Real Inner Product Space, $|s_{n}|^{2}=t_{n}$ for every $n$. Since $\bigl||s_{n}|-|z|\bigr|\le|s_{n}-z|$ by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity (ii), the sequence $(|s_{n}|)$ converges to $|z|$, hence $(|s_{n}|^{2})$ converges to $|z|^{2}$ by claim 2 of Arithmetic of Limits of Real Sequences; and $(t_{n})$ converges to $\sum_{k=1}^{\infty}c_{k}^{2}$. By uniqueness of limits, $|z|^{2}=\sum_{k=1}^{\infty}c_{k}^{2}$.

Finally fix $j\in\mathbb{N}$. By claim 3 of Inner Products Against Finite Sums, and Orthonormal Families, in a Real Inner Product Space, $\langle s_{n},e_{j}\rangle=c_{j}$ whenever $j\le n$. Given a positive $\varepsilon$, choose $N$ with $j\le N$ and $|s_{n}-z|<\varepsilon$ for $N\le n$; then, by The Cauchy–Schwarz Inequality in a Real Inner Product Space and $|e_{j}|=1$,

$$|c_{j}-\langle z,e_{j}\rangle|=|\langle s_{n}-z,e_{j}\rangle|\le|s_{n}-z|<\varepsilon .$$

As $\varepsilon$ was arbitrary, Comparison of Real Numbers with Arbitrary Positive Slack gives $|c_{j}-\langle z,e_{j}\rangle|=0$, so $\langle z,e_{j}\rangle=c_{j}$ by claim 1 of Properties of the Absolute Value in an Ordered Field.

Claim 3. Put $c_{k}=\langle x,e_{k}\rangle$. By claim 1 the series $\sum_{k=1}^{\infty}c_{k}^{2}$ converges, so by claim 2 the series $\sum_{k=1}^{\infty}c_{k}e_{k}$ converges, with sum $z$ say, and $\langle z,e_{j}\rangle=c_{j}=\langle x,e_{j}\rangle$ for every $j$. Hence $\langle x-z,e_{j}\rangle=0$ for every $j$ by Elementary Identities in a Real Inner Product Space (i), and since $(e_{k})$ is an orthonormal basis, $x-z=0_{H}$, that is, $x=z$.

Claim 4. The first identity is the norm identity of claim 2 with $c_{k}=\langle x,e_{k}\rangle$ and, by claim 3, $z=x$. For the second, claim 3 says that $\sum_{k=1}^{\infty}\langle x,e_{k}\rangle e_{k}$ converges with sum $x$, so claim 7 of Elementary Properties of Series in a Real Inner Product Space shows that the series with terms $\langle\langle x,e_{k}\rangle e_{k},y\rangle$ converges with sum $\langle x,y\rangle$. By condition (c) of Real Inner Product Spaces (i) and symmetry, $\langle\langle x,e_{k}\rangle e_{k},y\rangle=\langle x,e_{k}\rangle\langle e_{k},y\rangle=\langle x,e_{k}\rangle\langle y,e_{k}\rangle$.

Claim 5. (a) implies (b) by claim 3, and (b) implies (c) by the norm identity of claim 2.

(c) implies (a): if $\langle x,e_{k}\rangle=0$ for every $k$, then every partial sum of $\sum_{k=1}^{\infty}\langle x,e_{k}\rangle^{2}$ vanishes, by claim 3 of Properties of Finite Sums with the factor $0$, so the sum is $0$ and (c) gives $|x|^{2}=0$; hence $\langle x,x\rangle=0$ and $x=0_{H}$ by condition (d) of Real Inner Product Spaces (i).

(b) implies (d): given $x\in H$ and a positive $\varepsilon$, the partial sums $\sum_{k=1}^{n}\langle x,e_{k}\rangle e_{k}$ lie in $\operatorname{span}(e^{(n)})$ by the definition of the span and converge to $x$, so some element of the union lies within $\varepsilon$ of $x$. Hence every open ball about $x$ meets the union, and $x$ lies in its closure by Characterization of the Closure in a Metric Space by Open Balls. As $x$ was arbitrary, the union is dense.

(d) implies (a): suppose $\langle x,e_{k}\rangle=0$ for every $k$ and let $v\in\operatorname{span}(e^{(n)})$ for some $n$, say $v=\sum_{k=1}^{n}\lambda_{k}e_{k}$. By claim 2 of Inner Products Against Finite Sums, and Orthonormal Families, in a Real Inner Product Space, $\langle x,v\rangle=\sum_{k=1}^{n}\lambda_{k}\langle x,e_{k}\rangle=0$. Suppose $x\ne0_{H}$, so that $|x|$ is positive. Let $\varepsilon$ be positive. By (d) and Characterization of the Closure in a Metric Space by Open Balls there is such a $v$ with $|x-v|<\tfrac{\varepsilon}{|x|}$. Then, by Elementary Identities in a Real Inner Product Space (i) and The Cauchy–Schwarz Inequality in a Real Inner Product Space,

$$|x|^{2}=\langle x,x\rangle=\langle x,x-v\rangle+\langle x,v\rangle=\langle x,x-v\rangle\le|x|\,|x-v|<\varepsilon .$$

Since $\varepsilon$ was arbitrary, Comparison of Real Numbers with Arbitrary Positive Slack gives $|x|^{2}\le0$, contradicting the positivity of $|x|^{2}$. Hence $x=0_{H}$, and $(e_{k})$ is an orthonormal basis.