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, 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
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
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