Geodesic
On symmetric spaces with convergence in measure on reflexive subspaces
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 8 (2018), pp. 3-11

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A closed subspace $H$ of a symmetric space $X$ on $[0,1]$ is said to be strongly embedded in $X$ if in $H$ a convergence in $X$-norm is equivalent to the convergence in Lebesgue measure. We study symmetric spaces $X$ with the property that all their reflexive subspaces are strongly embedded in $X$. We prove that it is the case for all spaces, which satisfy an analog of the classical Dunford–Pettis theorem of relatively weakly compact subsets in $L_1$. At the same time the converse assertion fails for a wide class of separable Marcinkiewicz spaces.
Keywords: symmetric spaces, reflexive subspace, equicontinuity of norms.
Mots-clés : Marcinkiewicz space 
@article{IVM_2018_8_a0,
     author = {S. V. Astashkin and S. I. Strakhov},
     title = {On symmetric spaces with convergence in measure on reflexive subspaces},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {3--11},
     publisher = {mathdoc},
     number = {8},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2018_8_a0/}
}
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S. V. Astashkin; S. I. Strakhov. On symmetric spaces with convergence in measure on reflexive subspaces. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 8 (2018), pp. 3-11. http://geodesic.mathdoc.fr/item/IVM_2018_8_a0/