= Eigenvalue counting function
{title2=$N(\Lambda)=\#\{j:\lambda_j\leq\Lambda\}$}
For a nonnegative <self-adjoint operator> with <compact resolvent>, this function counts <eigenvalues> up to a given level with their multiplicities. For the <positive Laplace-Beltrami operator> on a closed $n$-dimensional <Riemannian manifold>, $N(\Lambda)\leq C(1+\Lambda)^{n/2}$. To prove this bound, multiply each vector in the low-eigenvalue subspace by a finite set of chart cutoffs and record its local <Fourier coefficients> of frequency at most $R$. The total number of retained coefficients is $O(R^n)$, while the sum of discarded squared norms is at most $CR^{-2}(1+\Lambda)\|u\|_2^2$ by <Parseval identity> and the energy bound. For $R$ a sufficiently large multiple of $\sqrt{1+\Lambda}$, vanishing of all recorded coefficients forces $u=0$. The recording map is injective and proves the dimension bound. Inverting this estimate gives $\lambda_j\geq c(j+1)^{2/n}-1$.
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