Proof of Finite Linear Combinations of Continuous Periodic Functions and Their Classes on the Torus

Each result cited is universally quantified over the data in its own statement. Throughout, $S$ is the successor map of Natural Numbers, so that $M+1=S(M)$; the exponent of Continuous Periodic Functions are Power-Integrable and Dense on the Torus is taken to be $p=2$ throughout.

We prove claims 1 and 2 together by induction. Let $T$ be the set of those $M\in\mathbb{N}$ for which the following holds: for every map $c:[M]\to\mathbb{R}$ and every choice of $g_{j}\in C_{\mathrm{per}}$ for $j\in[M]$, the map $u(x)=\sum_{j=1}^{M}c_{j}g_{j}(x)$ lies in $C_{\mathrm{per}}$ and satisfies $[\,u|_{Q}\,]=\sum_{j=1}^{M}c_{j}[\,g_{j}|_{Q}\,]$ in $L^{2}(\mathbb{T}^{n})$. In the latter identity every class $[\,g_{j}|_{Q}\,]$ is defined, since $g_{j}|_{Q}\in\mathcal{L}^{2}(\mathbb{T}^{n})$ by Continuous Periodic Functions are Power-Integrable and Dense on the Torus §member.

Base case. Let $M=1$ and let $c$ and $g_{1}$ be given. By claim 1 of Properties of Finite Sums, $u(x)=c_{1}g_{1}(x)$ for every $x$, so $u=c_{1}g_{1}$ is the scalar multiple, which lies in $C_{\mathrm{per}}$ by Elementary Properties of Lattice-Periodic Functions §algebra. By Continuous Periodic Functions are Power-Integrable and Dense on the Torus §linear, $[\,(c_{1}g_{1})|_{Q}\,]=c_{1}[\,g_{1}|_{Q}\,]$, and this equals $\sum_{j=1}^{1}c_{j}[\,g_{j}|_{Q}\,]$ by claim 1 of Properties of Finite Sums of Vectors. So $1\in T$.

Inductive step. Let $M\in T$, and let $c:[M+1]\to\mathbb{R}$ and $g_{j}\in C_{\mathrm{per}}$ for $j\in[M+1]$ be given. Let $c'$ be the restriction of $c$ to $[M]$ and let $v:\mathbb{R}^{n}\to\mathbb{R}$ be the map $v(x)=\sum_{j=1}^{M}c'_{j}g_{j}(x)$. By the restriction and recursion clause, claim 1 of Properties of Finite Sums, for every $x\in\mathbb{R}^{n}$

so $u=v+c_{M+1}g_{M+1}$ is the pointwise sum of $v$ and the scalar multiple $c_{M+1}g_{M+1}$. Since $M\in T$, the map $v$ lies in $C_{\mathrm{per}}$ and $[\,v|_{Q}\,]=\sum_{j=1}^{M}c'_{j}[\,g_{j}|_{Q}\,]$. The map $c_{M+1}g_{M+1}$ lies in $C_{\mathrm{per}}$ by Elementary Properties of Lattice-Periodic Functions §algebra, and so does the sum $u$, by the same clause. By Continuous Periodic Functions are Power-Integrable and Dense on the Torus §linear, applied to the sum $v+c_{M+1}g_{M+1}$ and to the scalar multiple $c_{M+1}g_{M+1}$,

the last equality by the restriction and recursion clause, claim 1 of Properties of Finite Sums of Vectors, applied in the vector space $L^{2}(\mathbb{T}^{n})$ to the $(M+1)$-tuple with components $c_{j}[\,g_{j}|_{Q}\,]$. Hence $M+1\in T$.

By Principle of Induction for the Natural Numbers, $T=\mathbb{N}$. This proves claim 1, the membership $u|_{Q}\in\mathcal{L}^{2}(\mathbb{T}^{n})$ following from Continuous Periodic Functions are Power-Integrable and Dense on the Torus §member, and claim 2.

Claim 3. Let $U\in L^{2}(\mathbb{T}^{n})$. The space $L^{2}(\mathbb{T}^{n})$ with $\langle\,\cdot\,,\cdot\,\rangle_{L^{2}}$ is a real inner product space by The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product. Let $G\in L^{2}(\mathbb{T}^{n})^{M}$ be the $M$-tuple with components $G_{j}=[\,g_{j}|_{Q}\,]$. By claim 2, $[\,u|_{Q}\,]=\sum_{j=1}^{M}c_{j}G_{j}$, so the second identity of Inner Products Against Finite Sums, and Orthonormal Families, in a Real Inner Product Space §combinations, applied with $w=U$, gives