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Let $(a_{mn})_{m,n\in\mathbb{N}}$ and $(b_m)$ be sequences of complex numbers.We say that $(a_{mn})$ and $(b_m)$ constitute an infinite system of linear equations in infinitely many variables if we seek a sequence $(x_n)$ of complex numbers such that $\forall m\in\mathbb{N}:$ $\sum_{n=1}^{\infty}a_{mn}x_n=b_m$. Note that in general the order of summation matters.

I am sort of a undergraduate student with focus on number theory and have some background in functional analysis (2 semesters functional analysis, 1 semester non-linear functional analysis, 1 semester operator algebras, 2 semesters PDEs), so I am sort of a becoming number-theorist with bias for functional analysis :-) That is also why I am fascinated by the above defined object as a sort of natural extension of a practical problem from linear algebra.

We have never dealt with this type of objects and I wasn´t able to find much on google that I could start something with, maybe partly because I have searched in the wrong way. That is why I have a request if you could recommend some introductory literature focused on such infinite systems of linear equations in infinitely many unknowns over $\mathbb{C}$.

Thanks in advance!

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If you are allowing infinite linear combinations, what kid of convergence of the infinite sum are you demanding? My first impression is that your question is too broadly posed, but perhaps other people disagree. – Yemon Choi Aug 22 '10 at 2:07
@EFQ: but once again it seems like you are asking other people to do all the work in thinking up what might be meant by your question. In case I wasn't clear above: I strongly feel that MO should not be a place for questions of the form "tell me stuff" or "write me a Wikipedia/nLab entry" – Yemon Choi Aug 22 '10 at 2:26
Dan: Maybe he'd want it to be "square" (or whatever is the equivalent term for infinite matrices)? Wouldn't "infinitely many equations in finitely many variables" be underdetermined? – J. M. Aug 22 '10 at 2:49
"Tell me about infinite systems of linear or non-linear equations over $\mathbb{R}$ or $\mathbb{C}.$ I don't see a meaningful question here and am voting to close. – Victor Protsak Aug 22 '10 at 3:06
You may want to look at Dieudonne's "History of functional analysis", which describes the history of this field in terms of solving equations of the type your question asks about. Also, the texts you used in functional analysis almost surely discuss Riesz's theory of compact operators, the Fredholm alternative, and related topics. These are modern outgrowths of people's attempts to come to grips with the problem of solving $Ax = b$ in an infinite-dimensional context. If you don't see how or why, Dieudonne's book will help to explain it. – Emerton Aug 27 '10 at 3:14
up vote 10 down vote accepted

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 $$ x_n+\sum_{m=1}^{\infty}a_{nm}x_m=b_n,\quad\mbox{such that}\quad \sum_n\sum_m a_{nm}^2<\infty \quad\mbox{ and }\quad \sum_nb_n^2<\infty, $$ 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).

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Take a look at Section 6 of Chapter III of Banach's book, which gives a result in the theory of $F$-spaces. The title of the section in the English translation is "Systems of linear equations in infinitely many unknowns".

(By coincidence I was reading this recently, and I admit that that is part of the reason I voted to reopen.)

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I would recommend taking a look at Hardy's "Divergent Series" it has quite a lot of nice ideas, in particular, I recall seeing exactly that example of a system of infinite equations in infinite unknowns related to fourier series.

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