Linearised Calder\'on problem: Reconstruction and Lipschitz stability for infinite-dimensional spaces of unbounded perturbations

We investigate a linearised Calder'on problem in a two-dimensional bounded simply connected $C^{1,\alpha }$ domain $\mathrm{\Omega }$. After extending the linearised problem for $L^2(\mathrm{\Omega })$ perturbations, we orthogonally decompose $L^2(\Omega) = \oplus_{k=0}^\infty $\mathcal{H}$_k$ and prove Lipschitz stability on each of the infinite-dimensional ${\mathcal{H}}_k$ subspaces. In particular, ${\mathcal{H}}_0$ is the space of square-integrable harmonic perturbations. This appears to be the first Lipschitz stability result for infinite-dimensional spaces of perturbations in the context of the (linearised) Calder'on problem. Previous optimal estimates with respect to the operator norm of the data map have been of the logarithmic-type in infinite-dimensional settings. The remarkable improvement is enabled by using the Hilbert-Schmidt norm for the Neumann-to-Dirichlet boundary map and its Fr'echet derivative with respect to the conductivity coefficient. We also derive a direct reconstruction method that inductively yields the orthogonal projections of a general $L^2(\mathrm{\Omega })$ perturbation onto the ${\mathcal{H}}_k$ spaces, hence reconstructing any $L^2(\mathrm{\Omega })$ perturbation.