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Inductive limit topologies on Orlicz spaces. (English) Zbl 0756.46014

In two previous papers [Bull. Pol. Acad. Sci., Math. 34, 675-687 (1986; Zbl 0639.46033) and Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 29, 255-275 (1990)] the authors studied inclusions and equalities among Orlicz spaces. He found, for example, a set \(Y(\varphi)\) of Young functions with the property that \[ E^ \varphi=\bigcup_{\psi\in Y(\varphi)}E^ \psi=\bigcup_{\psi\in Y(\varphi)}L^ \psi.\tag{*} \] In the present paper, he shows that the appropriate limit topologies on \(E^ \varphi\) in (*) coincide with the norm topology on \(L^ \varphi\) restricted to \(E^ \varphi\).

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46A13 Spaces defined by inductive or projective limits (LB, LF, etc.)
54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)

Citations:

Zbl 0639.46033
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