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Given two differential operators say $D_1$ and $D_2$ is there any meaningful way to define distance between them, does there exist some metric $d(D_1,D_2)$ that satisfies all the necessary properties? If there isn't a simple/natural way to define distance then what restrictions would it take to think about distance between differential operators? I know in general the operator is unbounded, and (correct me if I'm wrong) that means you can't define a topology induced by the norm.

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Where exactly do your operators act? Let's assume $D_1, D_2: \Gamma(E) \to \Gamma(F)$, where $E,F \to X$ are smooth vector bundles over the compact Riemannian manifold $X$ and let's say both of order $\leq k$ for some $k \in \mathbb{N}$. Passing to the $L^2$-space, we obtain two unbounded operators $D_1, D_2: L^2(E) \to L^2(F)$. It is indeed true that it does not make sence, to define the distance between two unbounded operators in norm topology. However, these operators are densely defined on the Sobolev space $L^2_k(E)$ of order $k$ and as such define bounded linear operators $D_1, D_2: L^2_k(E) \to L^2(F)$. Consequently, you can calculate the distance in $\mathcal{L}(L^2_k(E),L^2(F))$, i.e. the space of such bounded operators. You can also view them as operators $D_1, D_2: L^2(E) \to L^2_{-k}(F)$ (or consider any other Fredholm extension if you like) and calculate the distance there.

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