A condition for locally uniform convexity of Orlicz spaces. (Chinese. English summary) Zbl 0587.46025
We discuss the local uniform convexity in Orlicz spaces \(L^*_ M(G)\) with respect to the Luxemburg norm \(\| \cdot \|_{(M)}\) and obtain the following result.
Theorem. The Orlicz space \(L^*_ M(G)\) with respect to the Luxemburg norm \(\| \cdot \|_{(M)}\) is locally uniformly convex iff M(u) satisfies the \(\Delta\) \({}_ 2\)-condition for large u and M(n) is strictly convex.
Theorem. The Orlicz space \(L^*_ M(G)\) with respect to the Luxemburg norm \(\| \cdot \|_{(M)}\) is locally uniformly convex iff M(u) satisfies the \(\Delta\) \({}_ 2\)-condition for large u and M(n) is strictly convex.
MSC:
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 46B25 | Classical Banach spaces in the general theory |
