= Solution
On the <torus> $\mathbb T^n=(\mathbb R/2\pi\mathbb Z)^n$, choose normalized <Haar measure>, so that $e_k(x)=e^{ik\cdot x}$, $k\in\mathbb Z^n$, is an <orthonormal basis> of $L^2$. A <periodic Sobolev space> is defined, for any real $s$, by
$$
H^s(\mathbb T^n)=\left\{u:\sum_{k\in\mathbb Z^n}(1+|k|^2)^s|\widehat u(k)|^2<\infty\right\},\qquad
\|u\|_{H^s}^2=\sum_k(1+|k|^2)^s|\widehat u(k)|^2.
$$
For negative $s$, the coefficients describe <distributions> rather than necessarily functions. For a nonnegative integer $s$, the weight is comparable to $\sum_{|\alpha|\leq s}|k^\alpha|^2$, so this <norm> is equivalent to the sum of the squared $L^2$ <norms> of all <weak derivatives> through order $s$. Fourier truncation proves that smooth <trigonometric polynomials> are dense. The $L^2$ pairing identifies the continuous dual of $H^s$ with $H^{-s}$, giving <Sobolev duality>.
Two Fourier estimates explain the usefulness of these spaces. If $s>m+n/2$, the <Cauchy-Schwarz inequality> gives
$$
\sum_k |k|^m|\widehat u(k)|
\leq\|u\|_{H^s}\left(\sum_k\frac{|k|^{2m}}{(1+|k|^2)^s}\right)^{1/2}<\infty.
$$
<Uniform convergence> of the differentiated <Fourier series> proves the <Sobolev embedding> $H^s\hookrightarrow C^m$. If $\delta>0$, a bounded subset of $H^{s+\delta}$ has uniformly small high-frequency tails in $H^s$:
$$
\sum_{|k|>R}(1+|k|^2)^s|\widehat u(k)|^2
\leq(1+R^2)^{-\delta}\|u\|_{H^{s+\delta}}^2.
$$
Finite-dimensional Fourier truncation therefore proves the compact embedding $H^{s+\delta}\hookrightarrow H^s$, the periodic version of <Rellich-Kondrachov compactness theorem>. In addition, for $s>n/2$ multiplication is continuous $H^s\times H^s\to H^s$, making it a <Sobolev algebra>. Indeed the weighted coefficient of a product is bounded by the sum of the convolutions of $\langle k\rangle^s|\widehat u(k)|$ with $|\widehat v(k)|$ and of $|\widehat u(k)|$ with $\langle k\rangle^s|\widehat v(k)|$; Cauchy-Schwarz puts unweighted coefficients in $\ell^1$, and the $\ell^1*\ell^2\to\ell^2$ case of <Young's convolution inequality> finishes the estimate.
For <eigenfunction expansions>, distinguish ellipticity from <self-adjointness>. An arbitrary non-self-adjoint elliptic operator need not have an <orthonormal basis> of <eigenfunctions>. A useful definite setting is
$$
L u=-\sum_{i,j}\partial_i(a_{ij}(x)\partial_j u)+V(x)u,
$$
with smooth real periodic coefficients, $a_{ij}=a_{ji}$, real $V$, and <uniform ellipticity>
$$
\sum_{i,j}a_{ij}(x)\xi_i\xi_j\geq c|\xi|^2\qquad(c>0).
$$
Take the <self-adjoint> realization in $L^2$ with domain $H^2$. Its construction and spectral properties can be seen from the associated form on $H^1$,
$$
b(u,v)=\int\left(\sum_{i,j}a_{ij}\partial_j u\,\overline{\partial_i v}+V u\overline v\right).
$$
For a sufficiently large constant $C$, $b_C=b+C(\cdot,\cdot)$ is a positive <coercive> form: $b_C(u,u)\geq c_1\|u\|_{H^1}^2$, and $L+C\geq I$. The <Lax-Milgram theorem> supplies the unique weak solution of $(L+C)u=f$ for $f\in H^{-1}$; the form is Hermitian, which gives symmetry of its inverse on $L^2$.
The regularity input is the <periodic elliptic estimate>
$$
\|u\|_{H^{s+2}}\leq C_s\bigl(\|Lu\|_{H^s}+\|u\|_{H^s}\bigr).
$$
Here is why an elliptic operator gains two <derivatives>. With coefficients frozen at a point, its principal symbol $\sum a_{ij}\xi_i\xi_j$ dominates $c|\xi|^2$, and the estimate follows by multiplying <Fourier coefficients> by that symbol. On a sufficiently small coordinate patch, the difference of the leading coefficients from these constants has arbitrarily small supremum <norm>. In the integer-order differentiated estimate, this absorbs the highest <derivatives> into the left side; <derivatives> falling on coefficients contribute only lower-order terms. A <partition of unity> reduces the variable-coefficient problem to such patches. Its <commutators> with $L$ have order one, controlled by
$$
\|u\|_{H^{s+1}}\leq\varepsilon\|u\|_{H^{s+2}}+C_\varepsilon\|u\|_{H^s},
$$
which follows by splitting low and high <Fourier modes>. Absorbing these terms gives the displayed estimate. For a weak $H^1$ solution, the same localization with difference quotients first bounds its second <weak derivatives> for $f\in L^2$; differentiating and repeating then gives $f\in H^s\Rightarrow u\in H^{s+2}$ for nonnegative integer $s$. Interpolation and duality extend the scale to real orders. In particular $(L+C)^{-1}:L^2\to H^2$ is bounded.
Let $R=(L+C)^{-1}$. <Compactness> of $H^2\hookrightarrow L^2$ makes $R$ a <compact operator>. Symmetry of the form makes $R$ <self-adjoint>, and positivity of $L+C$ makes $R$ positive and <injective>. Its range is dense because $\ker R^*=\ker R=0$. The <spectral theorem for compact self-adjoint operators> consequently gives an <orthonormal basis> $\phi_j$ with
$$
R\phi_j=\mu_j\phi_j,\quad\mu_j>0,\quad\mu_j\longrightarrow0,\qquad
L\phi_j=\lambda_j\phi_j,\quad\lambda_j=\mu_j^{-1}-C\longrightarrow+\infty.
$$
<Elliptic regularity> bootstraps every $\phi_j$ to a <smooth function>. Each <eigenvalue> has finite multiplicity. Thus \b[a <self-adjoint> uniformly elliptic second-order operator on the <torus> has a complete smooth <orthonormal> <eigenfunction expansion> with discrete <eigenvalues> tending to infinity].
Writing $u_j=(u,\phi_j)$, every $u\in L^2$ has $u=\sum_j u_j\phi_j$ in $L^2$, with $\|u\|_2^2=\sum_j|u_j|^2$. Iterating the elliptic estimate shows that the graph <norm> of $(L+C)^m$ is equivalent to the $H^{2m}$ <norm>. Interpolation and duality give the more general spectral description
$$
\boxed{\|u\|_{H^s}^2\asymp\sum_j(\lambda_j+C)^s|u_j|^2.}
$$
The comparison constants depend on the operator and $s$. In particular <smooth function>s have coefficients decaying faster than every power of $\lambda_j+C$, and their expansions converge in every <Sobolev norm> and hence in every $C^m$ <norm>.
This gives a practical way to solve elliptic equations. The equation $Lu=f$ is solvable if and only if $f_j=0$ whenever $\lambda_j=0$; then one may take $u_j=f_j/\lambda_j$ on the other <eigenspaces> and add an arbitrary <vector> in $\ker L$. The Sobolev description proves the two-derivative gain and the <Fredholm alternative>, rather than leaving them as formal <series> manipulations. Similarly the solution of the <heat equation> $\partial_tu+Lu=0$ is
$$
u(t)=\sum_j e^{-t\lambda_j}u_j(0)\phi_j.
$$
For every $t>0$ the exponential decay at high <eigenvalues> places this solution in every <Sobolev space>, hence it is smooth.
For the <Laplacian> $-\Delta$ the <eigenfunctions> are precisely $e^{ik\cdot x}$ with <eigenvalues> $|k|^2$. The number with $|k|^2\leq R$ is the number of lattice points in a radius-$\sqrt R$ ball, asymptotic to its volume by comparison with unit cubes. For the general <self-adjoint> operator above, bounds on its form above and below by positive multiples of $\|\nabla u\|_2^2$ plus constants, together with the <Courant–Fischer min-max principle>, imply $\lambda_j+C\asymp j^{2/n}$. Thus the growth reflects <dimension> and the two-derivative order. <Compactness> of the <torus> is essential: on $\mathbb R^n$, translated bumps prevent the compact embedding, and the <Laplacian> instead has continuous <spectrum> described by the <Fourier transform>.
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