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Mat. Zametki, 1977, Volume 21, Issue 5, Pages 615–626 (Mi mz7994)  

Theorem on a convergence condition in the spaces

S. V. Lapin

Kaluga Branch, N. È. Bauman Moscow Higher Technical School

Abstract: For a given $\varphi$-function $\varphi(u)$, a condition on a $\varphi$-function $\psi(u)$ is found such that it is necessary and sufficient for the following to hold: $f_n(x)\to f(x)$ and $\|f_n(x)\|_\psi\le M$ ($1,2,…$) where $M>0$ is an absolute constant, then $\|f_n(x)-f(x)\|_\varphi\to0$ ($n\to\infty$). An analogous condition for convergence in Orlicz spaces is obtained as a corollary.

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English version:
Mathematical Notes, 1977, 21:5, 346–352

Bibliographic databases:

UDC: 517.5
Received: 16.06.1976

Citation: S. V. Lapin, “Theorem on a convergence condition in the spaces”, Mat. Zametki, 21:5 (1977), 615–626; Math. Notes, 21:5 (1977), 346–352

Citation in format AMSBIB
\Bibitem{Lap77}
\by S.~V.~Lapin
\paper Theorem on a~convergence condition in the spaces
\jour Mat. Zametki
\yr 1977
\vol 21
\issue 5
\pages 615--626
\mathnet{http://mi.mathnet.ru/mz7994}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=493298}
\zmath{https://zbmath.org/?q=an:0398.41027|0365.41006}
\transl
\jour Math. Notes
\yr 1977
\vol 21
\issue 5
\pages 346--352
\crossref{https://doi.org/10.1007/BF01788230}


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