Abstract
For a given ϕ-function ϕ(u), a condition on a ϕ-function ψ(u) is found such that it is necessary and sufficient for the following to hold: if fn(x) → f(x) and ∥f n (x)∥ψ⩽M (n=1, 2, ...) where M>0 is an absolute constant, then ∥f n (x)−f(x)∥ϕ→0(n→∞). An analogous condition for convergence in Orlicz spaces is obtained as a corollary.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature cited
A. Zygmund, Trigonometric Series, Cambridge Univ. Press (1959).
N. Dunford and J. Schwartz, Linear Operators, Wiley-Interscience, New York (1958).
W. Matuszewska and W. Orlicz, “A note on the theory of s-normed spaces of ϕ-integrable functions,” Stud. Math.,21, No. 1, 107–115 (1961).
S. Mazur and W. Orlicz, “On some classes of linear spaces,” Stud. Math.,17, No. 1, 97–119 (1958).
I. P. Natanson, Theory of Functions of a Real Variable [in Russian], Nauka, Moscow (1974).
M. A. Krasnosel'skii and Ya. B. Rutitskii, Convex Functions and Orlicz Spaces [in Russian], Fizmatgiz, Moscow (1958).
Author information
Authors and Affiliations
Additional information
Translated from Matematicheskie Zametki, Vol. 21, No. 5, pp. 615–626, May, 1977.
The author thanks V. A. Skvortsov for his constant attention and guidance on this paper.
Rights and permissions
About this article
Cite this article
Lapin, S.V. Theorem on a convergence condition in the spaces. Mathematical Notes of the Academy of Sciences of the USSR 21, 346–352 (1977). https://doi.org/10.1007/BF01788230
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01788230