I am interested in learning more about what the (formal) implications are when transforming a $n$-dimensional function space (e.g., the space of all $\mathbb{R}^n\to\mathbb{R}$) to a higherdimensional one (say, $\mathbb{R}^m\to\mathbb{R}$, $m>n$); in particular, in terms of consistency or range conditions (is there a difference?).
A real example could be the relationship between the result of the X-ray transform and John's equation. A dummy example could involve the operator $\mathcal{D}$ such that $g(x,y):=(\mathcal{D}f)(x,y) = f(x)$ -- clearly, $\frac{\partial g}{\partial y}=0$ is the (only?) range/consistency condition in that case.
I am thus looking for very general results such as "if we map from a space of dimension $n$ to one of dimension $m$, we find $m-n$ consistency equations". Also, results about 'requiredness' and 'sufficiency' are of interest.
What may be useful literature suitable for a non-professional mathematician to get an overview of this topic and confirm some hypotheses that I derived from particular examples (such as the related question, Range conditions on a linear operator)