In this paper^{1} on the history of functional analysis, the author mentions the following example of an infinite system of linear equations in an infinite number of variables $c_i = A_{ij} x_j$:

\begin{align*} \begin{array}{ccccccccc} 1 & = & x_1 & + & x_2 & + & x_3 & + & \dots \\ 1 & = & & & x_2 & + & x_3 & + & \dots \\ 1 & = & & & & & x_3 & + & \dots \\ & \vdots & & & & & & & \ddots \end{array} \to \begin{bmatrix} 1 \\ 1 \\ 1 \\ \vdots \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 & \dots \\ & 1 & 1 & \dots \\ & & 1 & \dots \\ & & & \ddots \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ \vdots \end{bmatrix} \end{align*}

as an example of a system such that any finite truncation of the system down to an $n \times n$ system has a unique solution $x_1 = \dots = x_{n=1} = 0, x_n = 1$ but for which the full system has no solution.

This book^{2} has the following passage on systems such as this one:

The Hahn-Banach theorem arose from attempts to solve infinite systems of linear equations... The key to the solvability is determining "compatibility" of the system of equations. For example, the system $x + y = 2$ and $x + y = 4$ cannot be solved because it requires contradictory things and so are "incompatible". The first attempts to determine compatibility for infinite systems of linear equations extended known determinant and row-reduction techniques. It was a classical analysis - almost solve the problem in a finite situation, then take a limit. A fatal defect of these approaches was the need for the (very rare) convergence of infinite products."

and then mentions a theorem about these systems that motivates Hahn-Banach:

Theorem 7.10.1 shows that to solve a certain system of linear equations, it is necessary and sufficient that a continuity-type condition be satisfied.

Theorem 7.10.1 (The Functional Problem):Let $X$ be a normed space over $\mathbb{F} = \mathbb{R}$ or $\mathbb{C}$, let $\{x_s \ : \ s \in S \}$ and $\{ c_s \ : \ s \in S \}$ be sets of vectors and scalars, respectively. Then there is a continuous linear functional $f$ on $X$ such that $f(x_s) = c_s$ for each $s \in S$ iff there exists $K > 0$ such that \begin{equation} \left|\sum_{s \in S} a_s c_s \right| \leq K \left\| \sum_{s \in S} a_s x_S \right\| \tag{1}, \end{equation} for any choice of scalars $\{a_s \ : \ s \in S \}$ for which $a_s = 0$ for all but finitely many $s \in S$ ("almost all" the $a_s = 0$).Banach used the Hahn-Banach theorem to prove Theorem 7.10.1 but Theorem 7.10.1 implies the Hahn-Banach theorem: Assuming that Theorem 7.10.1 holds, let $\{ x_s \}$ be the vectors of a subspace $M$, let $f$ be a continuous linear functional on $M$; for each $s \in S$, let $c_s = f(x_s)$. Since $f$ is continuous, $(1)$ is satisfied and $f$ possesses a continuous extension to $X$.

My question is:

If you knew none of the theorems just mentioned, how would one begin from the system $c_i = A_{ij} x_j$ at the beginning of this post and think of setting up the conditions of theorem 7.10.1 as a way to test whether this system has a solution?

How does this test show the system has no solution?

How do we re-formulate this process as though we were applying the Hahn-Banach theorem?

Does anybody know of a reference for the

*classical analysis*of systems in terms of infinite products?

^{1}Neal L. Carothers: *A Brief History of Functional Analysis*.

^{2}Lawrence Narici, Edward Beckenstein: *Topological Vector Spaces*, 2nd Edition.