A repeated eigenvalue has no canonical eigenfunction basis

Assume the Axiom of Choice and Countable Choice. For a symmetric elliptic Dirichlet operator on a nonempty bounded open set $\mathrm{\Omega }$, fix a weak eigenvalue $\lambda$ with eigenspace $E_{\lambda }$ of dimension $m\ge 2$ (The $L^2$ operator associated with a symmetric elliptic form, Symmetric elliptic weak eigenpairs, Discrete spectrum of a symmetric elliptic Dirichlet operator). The subspace $E_{\lambda }$ and the orthogonal projection $P_{\lambda }$ onto it are intrinsic to $L$ (and to the shifted solution operator), but no particular orthonormal basis of $E_{\lambda }$ is: for every unitary $U$ of the $m$-dimensional Hilbert space $E_{\lambda }$ the family $\{U{e}_j\}$ is another orthonormal eigenbasis, and every nonzero $u\in E_{\lambda }$ is an eigenfunction. Consequently downstream statements must refer to $E_{\lambda }$, to its dimension (the multiplicity), or to $U$-invariant quantities, and never to "the" eigenfunctions of a repeated eigenvalue. This concerns only the non-canonical choice of basis, not the existence of a Hilbert basis asserted by the discrete spectral theorem.

Among the invariant objects are the eigenspace $E_{\lambda }$, the orthogonal projection $P_{\lambda }$ onto it, and the multiplicity $\dim E_{\lambda }$, which is finite by Discrete spectrum of a symmetric elliptic Dirichlet operator; the eigenfunctions themselves are $L^2$ classes, and no canonical representative of a class is selected either (Eigenvalues, eigenvectors, eigenspaces $E_{\lambda }(T)=\ker (T-\lambda I)$, and the spectrum ${\sigma }_F(T)$ of an endomorphism, Real and complex inner-product spaces and their induced length, Orthonormal families, complete orthonormal systems and Hilbert bases, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). The remark records a convention for downstream statements; it proves nothing beyond linear algebra inside the finite-dimensional space $E_{\lambda }$.