### Isometric stability property of Banach spaces.

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Let ϕ: ℝ → ℝ₊ ∪ 0 be an even convex continuous function with ϕ(0) = 0 and ϕ(u) > 0 for all u > 0 and let w be a weight function. u₀ and v₀ are defined by u₀ = supu: ϕ is linear on (0,u), v₀=supv: w is constant on (0,v) (where sup∅ = 0). We prove the following theorem. Theorem. Suppose that ${\Lambda}_{\varphi ,w}(0,\infty )$ (respectively, ${\Lambda}_{\varphi ,w}(0,1)$) is an order continuous Lorentz-Orlicz space. (1) ${\Lambda}_{\varphi ,w}$ has normal structure if and only if u₀ = 0 (respectively, ${\int}^{v\u2080}\varphi \left(u\u2080\right)\xb7w<2andu\u2080<\infty ).$(2) ${\Lambda}_{\varphi ,w}$ has weakly normal structure if and only if ${\int}_{0}^{v\u2080}\varphi \left(u\u2080\right)\xb7w<2$.

A characterization of isometries of complex Musielak-Orlicz spaces ${L}_{\Phi}$ is given. If ${L}_{\Phi}$ is not a Hilbert space and $U:{L}_{\Phi}\to {L}_{\Phi}$ is a surjective isometry, then there exist a regular set isomorphism τ from (T,Σ,μ) onto itself and a measurable function w such that U(f) = w ·(f ∘ τ) for all $f\in {L}_{\Phi}$. Isometries of real Nakano spaces, a particular case of Musielak-Orlicz spaces, are also studied.

In this article, we consider the (weak) drop property, weak property (a), and property (w) for closed convex sets. Here we give some relations between those properties. Particularly, we prove that C has (weak) property (a) if and only if the subdifferential mapping of Cº is (n-n) (respectively, (n-w)) upper semicontinuous and (weak) compact valued. This gives an extension of a theorem of Giles and the first author.

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