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| In [[mathematics]], '''Frölicher spaces''' extend the notions of [[calculus]] and [[smooth manifold]]s. They were introduced in 1982 by the [[mathematician]] [[Alfred Frölicher]].
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| ==Definition==
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| 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
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| :''f'' : ''X'' → '''R'''
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| in ''F'' and each curve
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| :''c'' : '''R''' → ''X''
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| in ''C'', the following axioms are satisfied:
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| # ''f'' in ''F'' if and only if for each ''γ'' in ''C'', ''f'' . ''γ'' in C<sup>∞</sup>('''R''', '''R''')
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| # ''c'' in ''C'' if and only if for each ''φ'' in ''F'', ''φ'' . ''c'' in C<sup>∞</sup>('''R''', '''R''')
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| Let ''A'' and ''B'' be two Frölicher spaces. A map
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| :''m'' : ''A'' → ''B''
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| is called ''smooth'' if for each smooth curve ''c'' in ''C''<sub>''A''</sub>, ''m''.''c'' is in ''C''<sub>''B''</sub>. Furthermore the space of all such smooth maps has itself the structure of a Frölicher space. The smooth functions on
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| ''
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| :''C<sup>∞</sup>(''A'', ''B'')
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| are the images of
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| :<math>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)</math>
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| == References ==
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| * {{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
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| [[Category:Smooth functions]]
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| [[Category:Structures on manifolds]]
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| {{mathanalysis-stub}}
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| {{DEFAULTSORT:Frolicher Space}}
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I am Ernestine and was born on 15 October 1977. My hobbies are Writing and Cheerleading.
Feel free to visit my blog - Fifa 15 coin generator, go to this website,