- #1
bolbteppa
- 309
- 41
In this paper 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 has the following passage on systems such as this one:
and then mentions a theorem about these systems that motivates Hahn-Banach:
My question is:
\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 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{align*}
|\sum_{s \in S} a_s c_s | \leq K || \sum_{s \in S} a_s x_S || \ \ \ \ (1),
\end{align*}
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##.
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{align*}
|\sum_{s \in S} a_s c_s | \leq K || \sum_{s \in S} a_s x_S || \ \ \ \ (1),
\end{align*}
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?