When the solution of an infinite system of linear algebraic equations exist and is bounded? I face an infinite system of linear algebraic equations. What conditions must the matrix $\bf A$ and the vector $\bf b$ satisfy for the existence of the bounded solution of the linear system
$$
{\bf A} {\bf x} = {\bf b}?
$$
Discussions Cramer's rule and Physics forum do not help.

The answer seems to be given in the well-known monograph "Approximate methods of higher analysis" by Kantorovich and Krylov (pp. 26-27).
The infinite system
$$
{\bf A}{\bf x} = {\bf b}
$$
is called


*

*regular if $$ \sum_{j = 1}^\infty |a_{i j}| < 1, $$

*fully regular if for a positive constant $\theta$, $$ \sum_{j = 1}^\infty |a_{i j}| \leq 1 - \theta < 1. $$
Here $a_{i j}$ are the elements of the matrix ${\bf A}$.


Denote by
$$
c_i = 1 - \sum_{j = 1}^\infty |a_{i j}|.
$$
Then, the following theorem holds.
Theorem. If the infinite system of linear equations
$$
{\bf A}{\bf x} = {\bf b}
$$
is regular, and the inequality
$$
|b_i| \leq K c_i
$$
holds for a constant $K > 0$, then the bounded solution
$$
|x_i| \leq K
$$
to the infinite system exists.
Remark If the infinite system is fully regular, and
$$
|b_i| \leq K \theta
$$
holds, then the bounded solution
$$
|x_i| \leq K
$$
to the infinite system exists.
