Frölicher space

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:

1. f in F if and only if for each γ in C, f . γ in C^∞(R, R)
2. 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 C_A, m.c is in C_B. 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

${\displaystyle S:F_{B}\times C_{A}\times \mathrm {C} ^{\infty }(\mathbf {R} ,\mathbf {R} )'\to \mathrm {Mor} (\mathrm {C} ^{\infty }(A,B),\mathbf {R} ):(f,c,\lambda )\mapsto S(f,c,\lambda ),\quad S(f,c,\lambda )(m):=\lambda (f\circ m\circ c)}$

References

• Kriegl, Andreas; Michor, Peter W. (1997), The convenient setting of global analysis, Mathematical Surveys and Monographs, 53, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0780-4, section 23