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On the strongest locally convex Lebesgue topology on Orlicz spaces. (English) Zbl 0912.46031

Summary: Let \(L^\varphi\) be an Orlicz space defined by an Orlicz function \(\varphi\) taking only finite values with \(\liminf_{u\to\infty} {\varphi(u)\over u}> 0\) (not necessarily convex) over a complete, \(\sigma\)-finite and atomless measure space and let \((L^\varphi)^\sim_n\) stand for the order continuous dual of \(L^\varphi\). Then the strongest locally convex Lebesgue topology \(\tau\) on \(L^\varphi\) (= the Mackey topology \(\tau(L^\varphi,(L^\varphi)^\sim_n))\) is equal to the restriction of the strongest Lebesgue topology \(\eta\) on \(L^{\overline\varphi}\), where \(\overline\varphi\) is the convex minorant of \(\varphi\) and \(\tau\) is generated by a family of norms defined by some convex Orlicz functions.

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46A80 Modular spaces
46A20 Duality theory for topological vector spaces
46A40 Ordered topological linear spaces, vector lattices
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References:

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