The Extended Courant Property

Let $\Omega \subset \mathbb{R}^2$ be a bounded domain with piecewise smooth boundary, or a compact Riemannian surface, with or without boundary. Let $\Delta$ be the Laplace–Beltrami operator, with either Dirichlet, Neumann, or no boundary condition, and let $\lambda_1 < \lambda_2 \leq \lambda_3 \leq \cdots$ be the eigenvalues, counted with multiplicity. For $n \geq 1$, let $\mathcal{L}_n(\Omega)$ be the vector space of linear combinations of eigenfunctions associated with the first $n$ eigenvalues, and let $\beta_0(w)$ denote the number of nodal domains of $w$. Extended Courant Property. Every nontrivial $w \in \mathcal{L}_n(\Omega)$ satisfies

$$\beta_0(w) \leq n.$$

The classical Courant nodal domain theorem gives the same bound for an individual eigenfunction, whereas this extension concerns arbitrary nontrivial linear combinations of the first $n$ eigenspaces. The supplied material does not establish whether this assertion is proved or remains open.