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
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.
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Izvestiya: Mathematics, 2004, 68:6, 1217–1227
MSC: 26B05, 46A20, 46A22, 46A30, 46F05, 46G05, 47A10, 58C20
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
\by O.~G.~Smolyanov, S.~A.~Shkarin
\paper Gateaux complex differentiability and continuity
\jour Izv. RAN. Ser. Mat.
\jour Izv. Math.
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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
B. O. Volkov, “Lévy Laplacians in Hida calculus and Malliavin calculus”, Proc. Steklov Inst. Math., 301 (2018), 11–24
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