Foundational temp formal power series

Foundational temp4 $\succ$ Foundational temp formal power series $\succ$ foundational_temp_formal_power_series

Guide

formal power series
Probably must come only after group/ring/etc.
(objects of study of abstract algebra)
Q: how far back can analysis be pushed?
note that all formal power series are differentiable

Idea: Abstract theory of sequences $(a_n)_n,(b_n)_n\in X^{\mathbb N}$ with main objects of interests being

In particular, consider for a strcutre $\langle M,\cdot\rangle$ where $\sum_{i=0}^\infty$ is somehow defined and $a_n,b_n,B_k^{n,m}\in M$

$T(b)_k^n:=\sum_{m=0}^\infty B_k^{n,m}\cdot b_m$

$B(a,b)_k=\sum_{n=0}^\infty a_n\cdot T_k^n(b)$

i.e.

$B(a,b):=\sum_{n=0}^\infty\sum_{m=0}^\infty B_k^{n,m}\cdot a_n \cdot b_m$


Sequel of

Foundational temp4

Foundational temp4