Proof of The Trigonometric System is an Orthonormal Basis of the Square-Integrable Space of the Torus

Each result cited is universally quantified over the data in its own statement. We use the notation of The Flat Torus: Standing Notation: the cell $Q$, the class $\mathcal{L}^{2}(\mathbb{T}^{n})$, the class map $[\,\cdot\,]$, the periodic class $C_{\mathrm{per}}$ and the restriction $u|_{Q}$. Write $\langle\,\cdot\,,\cdot\,\rangle$ and $\lVert\,\cdot\,\rVert$ for the inner product and norm of $L^{2}(\mathbb{T}^{n})$; by The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product this norm is the norm $\lVert\,\cdot\,\rVert_{L^{2}(\mathbb{T}^{n})}$ of The Flat Torus: Standing Notation §lebesgue, in which the bounds of Continuous Periodic Functions are Power-Integrable and Dense on the Torus are stated. The exponent of Continuous Periodic Functions are Power-Integrable and Dense on the Torus is $p=2$ throughout, and for $u\in C_{\mathrm{per}}$ and $N\in\mathbb{N}$, $\sigma_{N}u$ is the Fejer mean of order $N$ of $u$. For maps $v,w:\mathbb{R}^{n}\to\mathbb{R}$, $v-w$ denotes the pointwise sum $v+(-1)w$ of $v$ and the scalar multiple $(-1)w$, formed as in Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set. The space $L^{2}(\mathbb{T}^{n})$ is a vector space over $\mathbb{R}$, as recorded in The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space, and the identities of vector arithmetic used below, such as $P-R=-(R-P)$, are elementary consequences of the conditions of that definition and of Elementary Identities in a Vector Space, whose claims 1, 2 and 5 fix the zero vector $0_{L^{2}}$, the difference $P-R=P+(-R)$ and the identity $(-1)R=-R$.

Claim 1. Let $U\in L^{2}(\mathbb{T}^{n})$ satisfy $\langle U,E_{k}\rangle=0$ for every $k\in\mathbb{Z}^{n}$. We show $U=0_{L^{2}}$.

Step 1: $U$ is orthogonal to every Fejer mean. Let $u\in C_{\mathrm{per}}$ and $N\in\mathbb{N}$. By The Fejer Means of a Continuous Periodic Function on the Torus §pairing and the hypothesis, $\langle U,[\,(\sigma_{N}u)|_{Q}\,]\rangle$ is a sum over the nonempty finite set $[2N+1]^{n}$ of The Reproducing Identity for the Fejer Kernels of the Torus every one of whose summands is a product with the factor $\langle U,E_{k(a)}\rangle=0$, hence equals $0=0\cdot1$; so by claim 4 of Properties of a Sum over a Finite Index Set, applied with $\lambda=0$ to the constant map $1$, the sum is $0$ times a real number, hence $0$, that is,

Step 2: an estimate. Let $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$. By The Lebesgue Space of Power-Integrable Functions §space the class $U$ is $[f]$ for some $f\in\mathcal{L}^{2}(\mathbb{T}^{n})$, so Continuous Periodic Functions are Power-Integrable and Dense on the Torus §dense furnishes $u\in C_{\mathrm{per}}$ with $\lVert U-[\,u|_{Q}\,]\rVert\le\varepsilon$. By Uniform Convergence of the Fejer Means of a Continuous Periodic Function on the Torus §uniform, applied to $u$ and $\varepsilon$, there is $N\in\mathbb{N}$ with $|\sigma_{N}u(x)-u(x)|\le\varepsilon$ for every $x\in\mathbb{R}^{n}$. The maps $\sigma_{N}u$ and $(-1)u$ lie in $C_{\mathrm{per}}$ by The Fejer Means of a Continuous Periodic Function on the Torus §member and Elementary Properties of Lattice-Periodic Functions §algebra, hence so does $v=\sigma_{N}u-u$, by the same clause; and $|v(x)|\le\varepsilon$ for every $x$, so Continuous Periodic Functions are Power-Integrable and Dense on the Torus §member, with $M=\varepsilon$, gives $\lVert[\,v|_{Q}\,]\rVert\le\varepsilon$. Two applications of Continuous Periodic Functions are Power-Integrable and Dense on the Torus §linear, first to the sum of $\sigma_{N}u$ and $(-1)u$ and then to the scalar multiple $(-1)u$ with $c=-1$, give $[\,v|_{Q}\,]=[\,(\sigma_{N}u)|_{Q}\,]+(-1)[\,u|_{Q}\,]=[\,(\sigma_{N}u)|_{Q}\,]-[\,u|_{Q}\,]$.

Writing $P=[\,u|_{Q}\,]$ and $R=[\,(\sigma_{N}u)|_{Q}\,]$, one has $U=(U-P)+(P-R)+R$ in the vector space $L^{2}(\mathbb{T}^{n})$, so by the additivity of the inner product in its second argument, Elementary Identities in a Real Inner Product Space §bilinear, and Step 1,

the first equality being Real Inner Product Space §norm. By claim 3 of Properties of the Absolute Value in an Ordered Field, The Cauchy-Schwarz Inequality in a Real Inner Product Space and claim 5 of Elementary Arithmetic in an Ordered Field,

where $\lVert P-R\rVert=\lVert-(R-P)\rVert=\lVert R-P\rVert=\lVert[\,v|_{Q}\,]\rVert\le\varepsilon$ by Elementary Identities in a Real Inner Product Space §homogeneity, and $0\le\lVert U\rVert$ by Real Inner Product Space §norm. Adding, by claims 2 and 3 of Elementary Arithmetic in an Ordered Field,

Step 3: conclusion. Suppose, for a contradiction, that $U\ne0_{L^{2}}$. Then $0<\lVert U\rVert$ by Elementary Identities in a Real Inner Product Space §vanishing, so $\lVert U\rVert^{-1}$ exists and is positive by claim 7 of Elementary Order Arithmetic in an Ordered Field. Take $\varepsilon=\lVert U\rVert\cdot2^{-1}\cdot2^{-1}$, which is positive by claims 8 and 5 of that lemma. Multiplying the estimate of Step 2 by $\lVert U\rVert^{-1}$, claim 5 of Elementary Arithmetic in an Ordered Field gives $\lVert U\rVert\le2\varepsilon=\lVert U\rVert\cdot2^{-1}$, using $\lVert U\rVert^{-1}\lVert U\rVert^{2}=\lVert U\rVert$ and $2\cdot2^{-1}=1$. But $\lVert U\rVert\cdot2^{-1}<\lVert U\rVert$ by claim 8 of Elementary Order Arithmetic in an Ordered Field, so claim 2 of that lemma gives $\lVert U\rVert<\lVert U\rVert$, which is impossible, since $a<b$ means $a\le b$ together with $a\ne b$. Hence $U=0_{L^{2}}$.

Claim 2. Let $\kappa:\mathbb{N}\to\mathbb{Z}^{n}$ be a bijection.

Orthonormality. Let $i,j\in\mathbb{N}$ with $i\ne j$. Then $\kappa(i)\ne\kappa(j)$: otherwise the element $\kappa(i)$ of $\mathbb{Z}^{n}$ would have the two distinct preimages $i$ and $j$, contrary to Bijection of Sets. Hence $\langle E_{\kappa(i)},E_{\kappa(j)}\rangle=0$ by The Trigonometric System on the Torus is Orthonormal §orthonormal. Next let $i\in\mathbb{N}$. The same clause gives $\langle E_{\kappa(i)},E_{\kappa(i)}\rangle=1$, so $\lVert E_{\kappa(i)}\rVert$ is the unique nonnegative real number whose square is $1$ by Real Inner Product Space §norm; since $0\le1$ and $1\cdot1=1$, this number is $1$. Thus $(E_{\kappa(j)})_{j\in\mathbb{N}}$ is an orthonormal sequence in $L^{2}(\mathbb{T}^{n})$.

Trivial orthogonal complement. Let $x\in L^{2}(\mathbb{T}^{n})$ satisfy $\langle x,E_{\kappa(j)}\rangle=0$ for every $j\in\mathbb{N}$. Let $k\in\mathbb{Z}^{n}$. By Bijection of Sets there is $j\in\mathbb{N}$ with $\kappa(j)=k$, so $\langle x,E_{k}\rangle=0$. Hence $\langle x,E_{k}\rangle=0$ for every $k\in\mathbb{Z}^{n}$, and claim 1 gives $x=0_{L^{2}}$. By Orthonormal Basis of a Real Hilbert Space §basis the orthonormal sequence $(E_{\kappa(j)})_{j\in\mathbb{N}}$ is an orthonormal basis of $L^{2}(\mathbb{T}^{n})$.