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Hadamard well-posedness requires existence of a solution for every admissible datum, uniqueness, and continuous dependence of the solution on the datum in the specified normed spaces. An inverse problem is ill-posed if any of these requirements fails. The choice of spaces and norms is part of this assertion: continuity in one norm does not establish continuity in a stronger norm.

For a bounded linear operator $A$, existence requires $y\in \mathrm{ran}A$, and uniqueness requires $\ker A=\{0\}$. Even when both hold for exact data, inversion can be unstable. For example, if $\Vert x_N{\Vert }_X=1$ but $\Vert A{x}_N{\Vert }_Y\to 0$, data $A{x}_N$ approach zero while their solutions do not approach zero. Thus $A^{-1}$ on its range is not continuous. Perturbed data can also leave the range entirely, so an exact solution need not exist after measurement error. These are distinct mechanisms of ill-posedness; the inverse scattering problem below exhibits unstable continuation even on data for which an outgoing continuation exists.