3. Regularization

When the image ImA of a linear operator to be inverted is not closed, then the inverse A ^–1 , or the generalized inverse A ^† , is not defined everywhere on $\mathcal{Y}$ and is not continuous. This is the case, for example, with non-degenerate (or non-finite rank) compact operators, and it's easy to see that the number of conditions in the problem is infinite. Appropriate solution techniques are then required, but we must also see that a well-posed but severely ill-conditioned problem behaves in practice like an ill-posed problem and must be treated by the same techniques.

Whether in finite or infinite dimension, a regularizer of the equation