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

This article is cited in 2 scientific papers (total in 2 papers)

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
Received: 14.11.2003

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
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\paper Gateaux complex differentiability and continuity
\jour Izv. RAN. Ser. Mat.
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\vol 68
\issue 6
\pages 157--168
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\jour Izv. Math.
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\issue 6
\pages 1217--1227
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    Citing articles on Google Scholar: Russian citations, English citations
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    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  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    2. B. O. Volkov, “Lévy Laplacians in Hida calculus and Malliavin calculus”, Proc. Steklov Inst. Math., 301 (2018), 11–24  mathnet  crossref  crossref  isi  elib  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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