Admissible constraint classes and restricted identifiability

Let $X$ and $Y$ be real vector spaces, let $U\subseteq X$ be nonempty, and let $A:X\to Y$ be linear. Supply metrics $d_X$ on $X$ and $d_Y$ on $Y$; for a purely algebraic instance with no specified metrics, use the discrete metrics $d_X(x,x^{\mathrm{\prime }})=0$ if $x=x^{\mathrm{\prime }}$ and $1$ otherwise, and likewise on $Y$. These metrics require no compatibility with the vector operations, and no continuity of $A$ is assumed. The discrete triangle inequality holds because, if two endpoints differ, at least one of the two successive pairs differs. Give $U$ and its subsets their restricted metrics. Thus every pair of real vector spaces still admits this convention. The corresponding metric-space inverse-problem model has admissible truths $U$ and forward map $A{\mathrm{\mid }}_U$. An admissible constraint class is a nonempty subset $C\subseteq U$; its constrained forward map is the restriction $A{\mathrm{\mid }}_C:C\to Y$. The class $C$ is identifiable for $A$ if $A{\mathrm{\mid }}_C$ is injective, meaning $Ax=A{x}^{\mathrm{\prime }}$ for $x,x^{\mathrm{\prime }}\in C$ implies $x=x^{\mathrm{\prime }}$. Define the difference set and kernel by

$C-C:=\{x-x^{\mathrm{\prime }}:x,x^{\mathrm{\prime }}\in C\},\ker A:=\{x\in X:Ax=0_Y\}\mathrm{.}$

Forward maps and admissible truth classes

Linear maps between vector spaces

Kernel of a linear map

Christian Clason, Regularization of Inverse Problems, Chapter 3 §§3.1–3.2

Tristan van Leeuwen and Christoph Brune, 10 Lectures on Inverse Problems and Imaging, Lecture 1 §1.2