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 Zh. Mat. Fiz. Anal. Geom., 2011, Volume 7, Number 3, Pages 225–284 (Mi jmag514)

Infinite dimensional spaces and cartesian closedness

Paolo Giordano

Department of Mathematics, University of Vienna, Nordbergstr 15, 1090 Wien, Austria

Abstract: Infinite dimensional spaces frequently appear in physics; there are several approaches to obtain a good categorical framework for this type of space, and cartesian closedness of some category, embedding smooth manifolds, is one of the most requested condition. In the first part of the paper, we start from the failures presented by the classical Banach manifolds approach and we will review the most studied approaches focusing on cartesian closedness: the convenient setting, diffeology and synthetic differential geometry. In the second part of the paper, we present a general settings to obtain cartesian closedness. Using this approach, we can also easily obtain the possibility to extend manifolds using nilpotent infinitesimal points, without any need to have a background in formal logic.

Key words and phrases: infinite dimensional spaces of smooth mappings, diffelogy, synthetic differential geometry, cartesian closedness.

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Document Type: Article
MSC: 58Bxx, 53Z05, 58B25
Language: English

Citation: Paolo Giordano, “Infinite dimensional spaces and cartesian closedness”, Zh. Mat. Fiz. Anal. Geom., 7:3 (2011), 225–284

Citation in format AMSBIB
\Bibitem{Gio11}
\by Paolo Giordano
\paper Infinite dimensional spaces and cartesian closedness
\jour Zh. Mat. Fiz. Anal. Geom.
\yr 2011
\vol 7
\issue 3
\pages 225--284
\mathnet{http://mi.mathnet.ru/jmag514}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2918489}
\zmath{https://zbmath.org/?q=an:1237.58010}


• http://mi.mathnet.ru/eng/jmag514
• http://mi.mathnet.ru/eng/jmag/v7/i3/p225

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This publication is cited in the following articles:
1. Giordano P., Wu E., “Categorical Frameworks For Generalized Functions”, Arabian J. Math., 4:4, SI (2015), 301–328
2. Giordano P., Wu E., “Calculus in the Ring of Fermat Reals, Part i: Integral Calculus”, Adv. Math., 289 (2016), 888–927
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