> [!theorem] > > Let $X$ be a [[Compactness|compact]] [[Riemannian Manifold|Riemannian n-manifold]] without boundary. Suppose that the [[Closed Eigenvalue Problem]] has solutions > $ > 0 \le \lambda_1 \le \lambda_2 \le \cdots > $ > repeated based on their geometric multiplicity, with corresponding eigenvectors $\seq{e_j}$. For any subspace $\cm \subset L^2(X)$ with dimension $k - 1$, > $ > \inf_{f \perp \cm}\frac{D[f, f]}{\norm{f}_{L^2}^2} \le \lambda_k > $ > with equality if $\cm = \span\bracs{e_j: j < k}$. > > *Proof*. Let $\cn = \span\bracs{e_j: j \le k}$. Since $\dim \cm < \dim \cn$, the [[Orthogonal Projection|orthogonal projection]] $\proj_\cm$ has non-trivial kernel. Therefore there exists $f \in \cn$ such that $\proj_\cm(f) = 0$ ($f \perp \cm$) and $f \ne 0$. In this case, > $ > D[f, f] = \sum_{j = 1}^k\lambda_j \angles{f, e_j}^2 \le \lambda_k\norm{f}_{L^2}^2 > $ > If $\cm = \span\bracs{e_j: j < k}$, then the reverse inequality comes from [[Rayleigh's Theorem]].