Using Fixed point iterations for solving system of linear equations Given a system of $n$ linear equations $$ x_i=\sum_{k=1}^{n}a_{ik}x_k+b_i \quad i=1,2,...,n$$ I'd like to employ the fixed point iteration method to find $x_i$. The fixed point iteration define $$ x_i^{N+1}=\sum_{k=1}^{n}a_{ik}x_k^{N}+b_i \quad i=1,2,...,n$$ with $x_i^{0}=0$. It states that, given that given the conditions of Banach fixed point system is satisfied, then $x_i^{\infty}=x_i$
What conditions are to be imposed upon $\sum_{k=1}^{n}a_{ik}$ to satsify Banach fixed point theorem? In particular I'm very interested in the case where $n=\infty$, that is a system of infinite equations.
 A: Just a partial solution, but suppose you restrict to the set of everywhere non-negative vectors 
$$\mathbb{V} = \big\{(x_1,\, x_2,\, \ldots) ~\big|~ x_i \in \mathbb{R}^+ \cup \{\infty\}\big\}$$ 
(where $\mathbb{R}^+$ are the non-negative reals). Then there is a complete partial order on $\mathbb V$ given by
$$(x_1,\, x_2,\, \ldots) ~\sqsubseteq~ \left(x_1',\, x_2',\, \ldots\right) \qquad\text{iff}\qquad \forall\, i\colon x_i \leq x_i'~,$$
with least element $\vec{0} = (0,\, 0,\, \ldots)$.
If you now further restrict $a_{ij} \geq 0$ and $b_i \geq 0$ then the update from $\left(x_1^N,\, x_2^N,\, \ldots\right)$ to $\left(x_1^{N+1},\, x_2^{N+1},\, \ldots\right)$ (as you have described it) is a function $F\colon \mathbb{V} \rightarrow \mathbb{V}$ which is monotonic with respect to the complete partial order $\sqsubseteq$. Thus, by the Kleene Fixed-Point Theorem, the fixed-point iteration $F^n\left(\vec{0}\right)$ will converge to a (the least) fixed-point of $F$.
This is also true if the vector space $\mathbb V$ is infinite-dimensional.
There is also a connection between the Banach Contraction Principle and the Kleene Fixed-Point Theorem, see here.
