There is a class of results of the form "eigenvalues are continuous in matrices" (e.g., this question and its answers). There is an analogous question as to whether the eigenspaces of an $n \times n$ complex matrix are continuous in some manner. One might try to pose this as a question about a function to a Grassmannian $G(k,\mathbb C^n)$, but one runs across the difficulty that the dimension can change.
The closest thing to what I want was as follows. Let $U \subseteq \mathbb C^{n \times n}$ be such that the multiset of eigenvalues of matrices $A \in U$ decomposes at each $A$ as union of multisets $\lambda(A)$ and $\mu(A)$ sharing no entries, or in other words that the (continuous!) eigenvalue-extraction function $U \to \mathbb C^n/\Sigma_n$ factors through $\mathbb C^\ell/\Sigma_\ell \times \mathbb C^m/\Sigma_m$ for some partition $\ell + m = n$. The sum $V(A)$ of $\zeta$-eigenspaces for $\zeta \in \lambda(A)$ is then defined at each $A \in U$ and one might ask if the union of the collection of $\{A\} \times V(A)$, with the natural projection, is a closed subbundle of the product bundle $U \times \mathbb C^n$.
I believe I can show it is, using the contour-integral expression for projection onto a sum of eigenspaces, but it seems clear to me this is a known result and I wonder if my proof is too hard.
Does anyone know a citation?
