## 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

• Realizations and Evaluations, i.e. maps in $X^{\mathbb N}\to Z$ (e.g. formal power series to functions and their evaluations)
• Transformations $T$, i.e. maps in $X^{\mathbb N}\to Z^{\mathbb N}$ (Fourier transform on component level)
• Binary mappings $B$, i.e. maps in $X^{\mathbb N}\times Y^{\mathbb N}\to Z^{\mathbb N}$ (e.g. Cauchy product)

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.

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