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, f∘γ in C∞(R, R) c in C if and only if for each φ in F, φ∘c 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, m∘c 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
S : F B × C A × C ∞ ( R , R ) ′ → M o r ( C ∞ ( A , B ) , R ) : ( f , c , λ ) ↦ S ( f , c , λ ) , S ( f , c , λ ) ( m ) := λ ( f ∘ m ∘ c ) {\displaystyle S:F_{B}\times C_{A}\times \mathrm {C} ^{\infty }(\mathbf {R} ,\mathbf {R} )'\to \mathrm {Mor} (\mathrm {C} ^{\infty }(A,B),\mathbf {R} ):(f,c,\lambda )\mapsto S(f,c,\lambda ),\quad S(f,c,\lambda )(m):=\lambda (f\circ m\circ c)}
References Kriegl, Andreas; Michor, Peter W. (1997), The convenient setting of global analysis, Mathematical Surveys and Monographs, vol. 53, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0780-4, section 23