Theorem on a convergence condition in the spaces

S. V. Lapin

Matematičeskie zametki, Tome 21 (1977) no. 5, pp. 615-626

Résumé

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.