In mathematics, Frölicher spaces extend the notions of calculus and smooth manifolds. They were introduced in 1982 by the mathematician Alfred Frölicher.
Definition A Frölicher space consists of a non-empty set X together with a subset C of Hom(R, X) called the set of smooth curves, and a subset F of Hom(X, R) called the set of smooth real functions, such that for each real function
f : X → R
in F and each curve
c : R → X
in C, the following axioms are satisfied:
f in F if and only if for each γ in C, in C∞(R, R) c in C if and only if for each φ in F, in C∞(R, R)
Let A and B be two Frölicher spaces. A map
m : A → B
is called smooth if for each smooth curve c in CA, is in CB. Furthermore, the space of all such smooth maps has itself the structure of a Frölicher space. The smooth functions on
C∞(A, B)
are the images of
References , section 23