Frölicher space - Alchetron, The Free Social Encyclopedia

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 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

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 )

References

Frölicher space Wikipedia

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