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, Template:Math in C^∞(R, R)
2. c in C if and only if for each φ in F, Template:Math 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, Template:Math 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

$S:F_B\times C_A\times {\mathrm{C}}^{\mathrm{\infty }}(𝐑,𝐑{)}^{\prime }\to \mathrm{M}\mathrm{o}\mathrm{r}({\mathrm{C}}^{\mathrm{\infty }}(A,B),𝐑):(f,c,\lambda )\mapsto S(f,c,\lambda ),S(f,c,\lambda )(m):=\lambda (f\circ m\circ c)$

References

• Template:Citation, section 23

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