Geodesic
Remarks on mean convergence
Matematičeskie zametki, Tome 21 (1977) no. 6, pp. 807-816 
P. L. Ul'yanov. Remarks on mean convergence. Matematičeskie zametki, Tome 21 (1977) no. 6, pp. 807-816. http://geodesic.mathdoc.fr/item/MZM_1977_21_6_a7/
@article{MZM_1977_21_6_a7,
     author = {P. L. Ul'yanov},
     title = {Remarks on mean convergence},
     journal = {Matemati\v{c}eskie zametki},
     pages = {807--816},
     year = {1977},
     volume = {21},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1977_21_6_a7/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this article a necessary and sufficient condition is established for the validity of the Cauchy criterion in the class of sequences of functions $\{f_n\}$ of the set $\varphi(L)$ which are convergent with respect to the $\varphi$-distance.