## Banach spaces in an analytic model of synthetic differential geometry.(English)Zbl 0923.32024

In a previous paper, the authors give an analytic well adapted model of Synthetic Differential Geometry. In this paper, an embedding of the category of open sets of complex Banach spaces and holomorphic maps into this model is constructed and two essential properties are proved. Namely, that this embedding preserves products and that it is consistent with the differential calculus, in the sense that the intrinsic synthetic differential calculus in the topos corresponds to the classical constructions of the theory of infinite dimensional holomorphy. In doing so, some interesting problems in the theory of ideals of analytic functions of several complex variables developed by Cartan are founded and solved.

### MSC:

 32K99 Generalizations of analytic spaces 51K10 Synthetic differential geometry

### Keywords:

synthetic differential geometry; embedding; Banach spaces
Full Text:

### References:

 [1] Bruno O. On a property of ideals of differentiable functions, Journal of the Australian Mathematical Society ( 1986 ) MR 832531 | Zbl 0603.26010 · Zbl 0603.26010 [2] Cartan H. Ideaux de fonctions analytiques de n variables complexes , Annales de l’Ecole Normale , 3 e serie, 61 ( 1944 ). Numdam | MR 14472 | Zbl 0035.17103 · Zbl 0035.17103 [3] Cartan H. Ideaux et Modules de Fonctions Analytiques de variables complexes, Bulletin de la Soc. Math. de France , t. 78 ( 1950 ). Numdam | MR 36848 | Zbl 0038.23703 · Zbl 0038.23703 [4] Douady A. Le Probleme des Modules pour les Sous-Espaces Analytiques Compacts d’un Espace Analytique donne , Ann. Inst. Fourier 16 , 1 , Grenoble ( 1966 ). Numdam | MR 203082 | Zbl 0146.31103 · Zbl 0146.31103 [5] Dubuc E. J. C\infty - Schemes , American Journal of Mathematics , Vol 103 ( 4 )( 1981 ) Zbl 0483.58003 · Zbl 0483.58003 [6] Dubuc E. J. , Taubin G. Analytic Rings , Cahiers de Topologie et Geometrie Differentielle XXIV - 3 ( 1983 ). Numdam | MR 728632 | Zbl 0575.32004 · Zbl 0575.32004 [7] Dubuc E.J. , Zilber J. Analytic Models of Synthetic Differential Geometry , Cahiers de Topologie et Geometrie Differentielle , Vol XXXV - 1 ( 1994 ). Numdam | Zbl 0790.32009 · Zbl 0790.32009 [8] Mujica , J. Holomorphic Functions and Domains of Holomorphy in Finite and Infinite Dimensions , North Holland Mathematics Studies 120 ( 1986 ). MR 842435 | Zbl 0586.46040 · Zbl 0586.46040 [9] Zilber J. Local Analytic Rings , Cahiers de Topologie et Geometrie Dif ferentielle XXXI - 1 ( 1990 ). Numdam | MR 1060606 | Zbl 0705.32002 · Zbl 0705.32002
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