The following is from my textbook Essential Partial Differential Equations, by Griffiths, Dold, and Silvester:
Note that, in the following textbook quote, the authors have previously defined that $\mathcal{L}$ denotes a differential operator in the space variables and $\mathscr{F}$ denotes a statement about a boundary value problem (BVP). Therefore, a BVP can be written in the compact form $\mathcal{L}u = \mathscr{F}$.
An overview of the situation can be obtained by considering a generic linear BVP written in the notation introduced in Sect. 2.1, namely
$$\mathcal{L}u = \mathscr{F} \ \ \ (2.17)$$
Suppose that we now make a "small" change $\delta \mathscr{F}$ to the data $\mathscr{F}$ and we denote the subsequent change to the solution by $\delta u$. Thus,
$$\mathcal{L}(u + \delta u) = \mathscr{F} + \delta \mathscr{F} \ \ \ (2.18)$$
Then, since $\mathcal{L}$ is a linear operator, (2.17) may be subtracted from (2.18) to give
$\mathcal{L}(\delta u) = \delta \mathscr{F} \ \ \ (2.19)$
so we see that $\delta u$ satisfies the same BVP as $u$ with $\mathscr{F}$ replaced by $\delta \mathscr{F}$. We now get to a definition which is the crux of the issue.
Definition 2.10
(Well-posed BVP) A boundary value problem which has a unique solution that varies continuously with the initial and boundary data is said to be well posed. A problem that is not well posed is said to be ill posed.
In the context of (2.19) this means that $\delta u$ should be "small" whenever $\delta \mathscr{F}$ is "small" in the sense that there are norms$^3$ $||\cdot||_a$, $||\cdot||_b$ and a constant $C$ that does not depend on $u$, $\mathscr{F}$ or $\delta \mathscr{F}$ so that
$$||\delta u||_a \le C||\delta \mathscr{F}||_b \ \ \ (2.20)$$
which must hold for all admissible choices of $\delta \mathscr{F}$. By choosing $\delta \mathscr{F} = -\mathscr{F}$ we deduce that $\delta u = -u$ and (2.20) then implies that
$$||u||_a \le C||\mathscr{F}||_b \ \ \ (2.21)$$
$^3$ The examples of ill-posed problems that we shall give are clear cut without the need to specify precisely which norms are used.
First of all, we can see that the arbitrary norms are denoted as $||\cdot||_a$ and $||\cdot||_b$. Since the subscripts $a$ and $b$ of these norms are different, I'm guessing that the author is saying that these norms can, but do not necessarily have to be, defined to be the same type of norm? Hmm, that does not make sense to me: why would we be making a comparison, such as was done with (2.20), using differently defined norms?!? That doesn't seem mathematically sensible?
Lastly, I'm assuming we got $||u||_a \le C||\mathscr{F}||_b \ \ \ (2.21)$ because the constants $\delta$ in $||\delta u||_a \le C||\delta \mathscr{F}||_b \ \ \ (2.21)$ factor out and cancel out, such that $||\delta u||_a \le C||\delta \mathscr{F}||_b \ \ \ (2.21) = |\delta | ||u||_a \le C |\delta | ||\mathscr{F}||_b = ||u||_a \le C||\mathscr{F}||_b \ \ \ (2.21)$ by the properties of norms? Just want to be sure.
Thank you for any help.