Scope of (Petrov-)Galerkin methods (I acknowledge that this question is somewhat conceptual, as I'm not an expert in the subject.)
I would like to understand what the general consensus is in the numerical analysis community on when to refer to a truncation scheme as a Galerkin method.
To be more precise, let me elaborate a bit further at some level of abstractness. As usual I am given an inverse linear problem $Ax=b$ in the unknown $x$, where $A$ is a linear operator, say, on a Hilbert space $\mathcal{H}$ and $b$ is a datum in the range of $A$. I am aware that in the vast majority of Galerkin methods (finite elements, in the first place) $A$ is a differential operator, but I am also to understand that it is not the differential nature of $A$ that characterises per se a Galerkin method (I hope I am right in that!) As for declaring the ambient space to be a Hilbert space (and not, more generally, a Banach space), this is also a slight restriction that I am making here to formulate my questions.
Then, as customary, in order to find a solution $x$ to
\begin{equation}\tag{1}
\langle Ax,v\rangle=\langle b,v\rangle\qquad \forall v\in\mathcal{H}
\end{equation}
one looks for approximate solutions $x_n\in V_n$ to
\begin{equation}\tag{2}
\langle Ax_n,v_n\rangle =\langle b,v_n\rangle\qquad \forall v_n\in V_n\,,
\end{equation}
 having chosen finite-dimensional subspaces  $V_n\subset\mathcal{H}$ (just for simplicity: $\dim V_n=n$), where $\langle\cdot,\cdot\rangle$ is the scalar product in the Hilbert space. (I am omitting here the standard generalisation to the Petrov-Galerkin scheme in which the truncation spaces $V_n$ and $W_n$ are different.) Equation (2) is then re-written in matrix form
\begin{equation}\tag{3}
A_n x_n = b_n
\end{equation}
and then one studies this finite-dimensional problem. Furthermore, one controls the convergence of the error $x-x_n$ or the residual $Ax_n-b$, or other such indicators.
Now, I suppose that this framework is still too general and is not yet the Galerkin framework. If so, I would like to understand:


*

*Do we speak of Galerkin methods only when the operator $A$ has certain features, such as being symmetric, or coercive, or bounded, etc.?

*Suitable features on $A$ ensure (1) to have a unique solution, yet leaving room for (2) to have infinitely many solutions or none at all. Can one speak of Galerkin methods also in the latter case?


Suggestions for comprehensive references would also be most welcome!
 A: The idea under the Galeking method is to approximate the solution space $V$ with a family of space $V_n$ s.t.:
$$
\forall v \in V \; \inf_{v_h \in V_h} || v - v_h || \longrightarrow 0 \; \text{ for}\; h \longrightarrow 0
$$
and not the operators. Essentially this is the idea of Galeking method.
It can be express in a, little bit, more general form. For reference see [1] chapter 9, in particular 9.1.
So the hypothesis under the operator $A$ (attention ambiguous notation because generally $A$ is the matrix that emerge from the discretization of the bilinear form $a(\cdot , \cdot)$), do not characterize the method itself. Hypotheses are used to satisfy the theorem of Lax-Milligram (here in a more general form) to show the existence and uniqueness.
I hope I understand the question.

[1] Atkinson, Kendall; Han, Weimin, Theoretical numerical analysis. A functional analysis framework, Texts in Applied Mathematics 39. Berlin: Springer (ISBN 978-1-4419-0457-7/hbk; 978-1-4419-0458-4/ebook). xvi, 625 p. (2009). ZBL1181.47078.
