= Solution
<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>. \b[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 $y\in\operatorname{ran}A$, and uniqueness requires $\ker A=\{0\}$. Even when both hold for exact data, inversion can be unstable. For example, if $\|x_N\|_X=1$ but $\|Ax_N\|_Y\to0$, data $Ax_N$ 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.
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