Explaining the stability of a numerical scheme through functional analysis In this paper, the stability of a discretization $$L_hx_h=y~~~~~~~~~~~(*) $$on a continuous problem $$Lx=y~~~~~~~~~~~(**)$$ where $L\in BL(X,Y)$ and $h$ represents the mesh parameters, is given by the uniform boundedness of the inverse operator $L_h^{-1}$. That is, the numerical scheme $(*)$ is said to be stable if $$\lVert L_h^{-1} \lVert  \leq C,$$ for some $C>0$ irrespective of the family of mesh parameters represented by $h$.
My doubts:
How can we ensure the boundedness of the most often partial differential operators appear in $(*)$ as we have seen that derivative operators are not continuous?
How can we practically explain the connection between stability and the 'uniform boundedness'?
 A: One notion of stability is that "for small differences in input, we get small differences in output". Mathematically, it could be, e.g., for two pieces of data $y_1, y_2$ and their respective solutions $x_1,x_2$, you would want
$$\|x_1-x_2\| \leq C \|y_1-y_2\|.$$
If $y_1$ and $y_2$ are 'close' in some sense, then the solutions to the inverse problem should also be 'close', perhaps up to an extra multiplicative factor $C$. The uniform boundedness of $L_h^{-1}$ gives precisely this notion of stability, since (for this system) a solution is produced via $x=L_h^{-1}y$. So
$$\|x_1-x_2\|=\|L_h^{-1}y_1-L_h^{-1}y_2\|\leq\|L_h^{-1}\|\|y_1-y_2\|\leq C\|y_1-y_2\|.$$
Why is $L_h$ bounded?: Since (as far as I can tell) $L_h$ is acting on a finite dimensional discretization. Even though it represents differentiation, it is still bounded since it is acting on a finite dimensional space (and all linear operators in finite dimensional spaces are bounded).
Concerning $(**)$: this is a very good observation, and in general, a linear operator involving differentiation will not be bounded in an infinite-dimensional space (and its inverse may not even exist)! So very careful analysis must be done in order to ensure that, as the grid $h$ refines, you actually converge to a solution of the continuous problem. Check out the literature/proofs on Galerkin methods for more info on this :-)
