Geodesic
Theorem on a convergence condition in the spaces
Matematičeskie zametki, Tome 21 (1977) no. 5, pp. 615-626 
S. V. Lapin. Theorem on a convergence condition in the spaces. Matematičeskie zametki, Tome 21 (1977) no. 5, pp. 615-626. http://geodesic.mathdoc.fr/item/MZM_1977_21_5_a4/
@article{MZM_1977_21_5_a4,
     author = {S. V. Lapin},
     title = {Theorem on a~convergence condition in the spaces},
     journal = {Matemati\v{c}eskie zametki},
     pages = {615--626},
     year = {1977},
     volume = {21},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1977_21_5_a4/}
}
TY  - JOUR
AU  - S. V. Lapin
TI  - Theorem on a convergence condition in the spaces
JO  - Matematičeskie zametki
PY  - 1977
SP  - 615
EP  - 626
VL  - 21
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/MZM_1977_21_5_a4/
LA  - ru
ID  - MZM_1977_21_5_a4
ER  - 
%0 Journal Article
%A S. V. Lapin
%T Theorem on a convergence condition in the spaces
%J Matematičeskie zametki
%D 1977
%P 615-626
%V 21
%N 5
%U http://geodesic.mathdoc.fr/item/MZM_1977_21_5_a4/
%G ru
%F MZM_1977_21_5_a4

Voir la notice de l'article provenant de la source Math-Net.Ru

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 $\|f_n(x)\|_\psi\le M$ ($1,2,\dots$) 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.