(Schwartz, p.80). Thus the solution f continuously depends on F in these spaces providing well-posedness. If density f exists as a regular integrable function, its integral coincides with the function F and integration by parts of (6) provides (5) . Thus f , the solution to the inverse problem in the space D ∗ is consistent with the solution when it exists as an ordinary function. ■ Corollary. The result of the Theorem applies in the space of generalized functions on D$_{m}$ ( W ) , m ≥ k. Proof. Indeed, consider the space D$_{∞}$ ( W ) ⊂ D$_{k}$ ( W ) . By the theorem the in- verse problem provides the density function f defined as a linear continuous functional on D$_{∞}$ ( W ) via (5) . We can extend the functional f to D$_{k}$ ( W ) as a linear continuous functional. First note that since F is a regular locally integrable function it represents an element in D ∗ $_{k}$; then define the functional in D ∗ k by (5) for any ψ ∈ G$_{k}$, denote it ˜ f to distinguish from f defined on D$_{∞}$ ( W ) . This ˜ f represents a linear continuous functional, so an element in D ∗ $_{k}$. There is an injective mapping of linear topological spaces D ∗ k → D ∗ ∞ (Sobolev, 1992 ; in notation there C ( k )# → C ( ∞ $^{)#}$) , thus by this mapping ˜ f maps into f and the inverse problem is solved in D ∗ k and is well-posed there . ■ 12