TITLE: "Algebrization" of Analytic Concepts
QUESTION [5 upvotes]: Please excuse my inventing new words in the title. I've been studying algebraic geometry, and one of my favourite parts of the subject (at least, before schemes) is the way that one can describe, or even define, the tangent space at a point $P$ of a variety $V$ purely algebraically, as the (dual space of) $\mathfrak{m}/\mathfrak{m}^2$, where $\mathfrak{m}$ is the unique maximal ideal of the local ring $\mathcal{O}_{V,P}$.
Are there other instances of concepts which are traditionally analytic in nature, but have been given alternative algebraic descriptions?
My search yielded little beyond differential algebra, which seems to be an underdeveloped subject of study, but I would think that if one wanted to "re-imagine" analysis using abstract algebra, this would be a good place to start - taking the interesting algebraic properties of derivative (such as the product rule) and building them into the definition of some new algebraic object. Algebraic analysis is apparently also a subject of study, but information is scarce.
Conversely, are there instances of algebraic concepts being re-cast only in terms of analysis?

REPLY [2 votes]: Plenty of things can be done in algebra. Completion is typically an analytical thing but it can be done through algebra. Let $A$ be some algebraic structure and for all $i\in\Bbb N$ we have that $S_i$ is a substructure such that we can make a quotient $A/S_i$, or if we go more exotic we would deal with congruence relations, such that $S_i\supseteq S_{i+1}$. 
Anyhow using this we can deal with cauchy sequences by defining that, using additive notation, a sequence $(x_i)$ is cauchy if for a given $k$ there always exists an $N$ such that for $i,j>N$ we have that $x_i-x_j\in S_k$. Through this we can complete rings, groups, algebras, modules and much else. This is a simple way of getting the $p$-adic numbers. This is also reversable as the sequence of substructures induces a natural metric.
The derivative has been mentioned before is another.