fa.functional analysis - infinitely many linear equations in infinitely many variables

The systems of this kind are fairly common in applications. For example, they naturally appear when solving boundary value problems for linear partial differential equations using the method of separation of variables.

Predictably, the problem is not meaningful for any sequences {$a_{nm}$}, {$b_m$}, but only for sufficiently well-behaved ones. If, for example, you were to consider systems of the form
$$ x_n+sum_{m=1}^{infty}a_{nm}x_m=b_n,quadmbox{such that}quad sum_nsum_m a_{nm}^2<infty quadmbox{ and }quad sum_nb_n^2<infty, $$
then this system possesses a unique solution in the Hilbert space $l_2$ such that $sum_n x_n^2<infty$ (assuming that the problem is not singular, i.e. that $det(I+A)ne0$). These requirements are too restrictive for some applications, hence there is a body of literature concerned with various kinds of regularity conditions involving {$a_{nm}$} and {$b_m$}, weaker than above, which ensure the well-posedness of the problem and enable numerical solution of such systems (which is usually done by truncation; see the appropriate accuracy estimates in F. Ursell (1996) "Infinite systems of equations: the effect of truncation", Quarterly Journal of Mechanics and Applied Mathematics, 49(2), 217--233).

One good old book that discusses these systems in some detail was written by By L. V. Kantorovich and V. I. Krylov and is called "Approximate methods of higher analysis" (New York: Interscience Publishers, 1958).