Post by Charlie-BooWho has considered a system of general function manipulations where the primitive object is a function?
Calculus maps a continuous function to its derivative.
Any two functions f:A=>B and g:B=>C define the composition f(g).
C-B
It's called "function theory", where the elements are functions.
Pretty much "the theory of relations" happens to suffice, where function
theory gets a bit descriptive, while relations are among
other primitives. (Usually enough two-place relations, vis-a-vis
predicates as simply relations to the consequents of evaluating
truth-values.)
You kind of get into "operator calculus" this way, for whatever
operations in whatever space like composition, convolution, and
so on, vis-a-vis other elements under the operator calculus like
differentiation and integration, besides usual things like the
operations of arithmetic, and this kind of thing.
I suppose you found it about everywhere.
"Operator calculus" I think is what you're looking for.
There's sort of a dichotomy between the elements and the
spaces, complementary duals, inner and outer, all this
kind of thing.
So, algebras and abstract algebra kind of help, but, lots
of the relations are only semi- or just various kinds of
magmas, about elements, pair-wise their operators, their
"completions" where they exist, implicits and explicits,
"where" and whether "everywhere", and these kinds of things.
So, yeah it's usually called "analysis", "functional analysis".
arithmetic <- various operations
algebra <- quite various structures
function theory
topology
geometry
Any of these is sort of their own "theory", while,
also they sit altogether in one "theory".