en>Dithridge at 20:27, 11 November 2014

2014-11-11T20:27:47Z

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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]]~~.		I am Ernestine and was born on 15 October 1977. My hobbies are Writing and Cheerleading.<br><br>Feel free to visit my blog - Fifa 15 coin generator, [http://garinmahal.com/component/k2/item/7-meet-the-fastest-crawler-in-paratroopers go to this website],

	~~==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~~<~~sup~~>∞<~~/sup~~>~~('''R'''~~, ~~'''R''')~~
	~~# ''c'' in ''C'' if and only if for each ''φ'' in ''F'', ''φ''~~ . ~~''c'' in C<sup>∞<~~/~~sup>('''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''<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~~
	''
	~~:''C<sup>∞<~~/~~sup>(''A'', ''B'')~~

	~~are~~ the ~~images of~~
	:<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>

	~~== References ==~~
	* {{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~~

	~~[[Category:Smooth functions]]~~
	~~[[Category:Structures on manifolds]]~~

	~~{{mathanalysis-stub}}~~
	~~{{DEFAULTSORT:Frolicher Space}}~~

en>Reaper Eternal: Reverted edits by 86.112.110.166 (talk) to last version by Melcombe

2012-03-21T17:37:34Z

Reverted edits by 86.112.110.166 (talk) to last version by Melcombe

New page

In [[mathematics]], '''Frölicher spaces''' extend the notions of [[calculus]] and [[smooth manifold]]s. 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
:<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>

== References ==
* {{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

[[Category:Smooth functions]]
[[Category:Structures on manifolds]]

{{mathanalysis-stub}}
{{DEFAULTSORT:Frolicher Space}}

Factored language model - Revision history

en>Dithridge at 20:27, 11 November 2014

en>Reaper Eternal: Reverted edits by 86.112.110.166 (talk) to last version by Melcombe