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 Izv. RAN. Ser. Mat., 2004, Volume 68, Issue 6, Pages 157–168 (Mi izv517)

Gateaux complex differentiability and continuity

O. G. Smolyanov, S. A. Shkarin

M. V. Lomonosov Moscow State University

Abstract: As is known, there are everywhere discontinuous infinitely Fréchet differentiable functions on the real locally convex spaces $\mathcal D(\mathbb R)$ and $\mathcal D'(\mathbb R)$ of finitely supported infinitely differentiable functions and, respectively, of generalized functions. In this paper the relationship between the complex differentiability and continuity of a function on a complex locally convex space is considered. We describe a class of complex locally convex spaces, which includes the complex space $\mathcal D'(\mathbb R)$, such that every Gateaux complex-differentiable function on a space of this class is continuous. We also describe another class of locally convex spaces, which includes the complex space $\mathcal D(\mathbb R)$, such that on every space of this class there is an everywhere discontinuous infinitely Fréchet complex-differentiable function whose derivatives are continuous.

DOI: https://doi.org/10.4213/im517

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English version:
Izvestiya: Mathematics, 2004, 68:6, 1217–1227

Bibliographic databases:

UDC: 517.98
MSC: 26B05, 46A20, 46A22, 46A30, 46F05, 46G05, 47A10, 58C20

Citation: O. G. Smolyanov, S. A. Shkarin, “Gateaux complex differentiability and continuity”, Izv. RAN. Ser. Mat., 68:6 (2004), 157–168; Izv. Math., 68:6 (2004), 1217–1227

Citation in format AMSBIB
\Bibitem{SmoShk04} \by O.~G.~Smolyanov, S.~A.~Shkarin \paper Gateaux complex differentiability and continuity \jour Izv. RAN. Ser. Mat. \yr 2004 \vol 68 \issue 6 \pages 157--168 \mathnet{http://mi.mathnet.ru/izv517} \crossref{https://doi.org/10.4213/im517} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2108527} \zmath{https://zbmath.org/?q=an:1082.46031} \transl \jour Izv. Math. \yr 2004 \vol 68 \issue 6 \pages 1217--1227 \crossref{https://doi.org/10.1070/IM2004v068n06ABEH000517} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000227279000008} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33746512634} 

• http://mi.mathnet.ru/eng/izv517
• https://doi.org/10.4213/im517
• http://mi.mathnet.ru/eng/izv/v68/i6/p157

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Smolyanov O.G., “Solutions of D. A. Raikov's problems in the theory of topological vector spaces”, Russ. J. Math. Phys., 15:4 (2008), 522–529
2. B. O. Volkov, “Lévy Laplacians in Hida calculus and Malliavin calculus”, Proc. Steklov Inst. Math., 301 (2018), 11–24
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