In this paper1 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 book2 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:
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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?
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How does this test show the system has no solution?
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How do we re-formulate this process as though we were applying the Hahn-Banach theorem?
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Does anybody know of a reference for the classical analysis of systems in terms of infinite products?
1Neal L. Carothers: A Brief History of Functional Analysis.
2Lawrence Narici, Edward Beckenstein: Topological Vector Spaces, 2nd Edition.
Best Answer
I will try to provide some rather incomplete answers to your questions.
The above result also implies Hahn Banach as the author says.