mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, Frölicher spaces extend the notions of

calculus Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...

and

smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may ...

s. They were introduced in 1982 by the

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

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'', in C^∞(R, R) # ''c'' in ''C'' if and only if for each ''φ'' in ''F'', 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'', 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^(\mathbf, \mathbf)' \to \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)

References

* , section 23 Smooth functions Structures on manifolds {{mathanalysis-stub