Infinite-Dimensional Manifolds as Ringed Spaces

被引:1
作者
Egeileh, Michel [1 ]
Wurzbacher, Tilman [2 ,3 ]
机构
[1] St Josephs Univ, Fac Sci, BP 11-514 Riad El Solh, Beirut 11072050, Lebanon
[2] Univ Lorraine, Inst Elie Cartan Lorraine, F-57045 Metz, France
[3] CNRS, F-57045 Metz, France
关键词
Infinite-dimensional manifolds; ringed spaces; smoothness of maps;
D O I
10.4171/PRIMS/53-1-6
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We analyze the possibility of defining infinite-dimensional manifolds as ringed spaces. More precisely, we consider three definitions of manifolds modeled on locally convex spaces: in terms of charts and atlases, in terms of ringed spaces and in terms of functored spaces, as introduced by Douady in his thesis. It is shown that for large classes of locally convex model spaces (containing Frechet spaces and duals of Frechet Schwartz spaces), the three definitions are actually equivalent. The equivalence of the definition via charts with the definition via ringed spaces is based on the fact that for the classes of model spaces under consideration, smoothness of maps turns out to be equivalent to their scalarwise smoothness (i.e., the smoothness of their composition with smooth real-valued functions).
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页码:187 / 209
页数:23
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