yawiki.org entry for Frölicher space

	Navigation Home Recent Most Active Popular Credits RSS

	Help How to Edit Help

[Edit]

In mathematics, Frölicher spaces extend the notions of calculus and smooth manifolds. They were introduced in 1982 by the mathematician Alfred Frölicher.

Frölicher space
Definition

top

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

and each curve

c
R → X

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 imes C_A imes mathrm^(mathbf, mathbf)' o mathrm(mathrm^(A, B), mathbf)
(f, c, lambda) mapsto S(f, c, lambda), quad S(f, c, lambda)(m)
= lambda(f circ m circ c)

-->

	Source Privacy License Download Contact Us Atlas Scientus.org Dictionary (Yet Another Wiki) RC : 1.41 MIT OpenCourseWare This article is licensed under the GNU Free Documentation License [copyleft]. It uses material from the Wikipedia article "Frölicher space". link