2. Linear problems

In this section, we study the main properties of linear inverse problems. We place ourselves in the framework of Hilbert spaces, so that the results apply (for example) to integral equations of the first kind, but we indicate the simplifications that come into play in finite dimension. We then introduce the fundamental tool of singular value decomposition. Finally, we show how singular value decomposition can be used to analyze least-squares problems.

Throughout this section, we will refer to A as a continuous linear operator from a Hilbert space E into a Hilbert space $F\mathrm{\hspace{0.17em}}\mathrm{:}\mathrm{\hspace{0.17em}}A\in \mathfrak{L}(E,$