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:

f in F if and only if for each γ in C, {{math|f∘γ}} in C^∞(R, R)
c in C if and only if for each φ in F, {{math|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, {{math|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

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

References

{{Citation | last1=Kriegl | first1=Andreas | last2=Michor | first2=Peter W. | title=The convenient setting of global analysis | publisher=American Mathematical Society | location=Providence, R.I. | series=Mathematical Surveys and Monographs | isbn=978-0-8218-0780-4 | year=1997 | volume=53}}, section 23

Category:Smooth functions

Category:Structures on manifolds