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In this paper we investigate Orlicz sequence spaces with regard to certain geometric properties that have proved to be important in fixed point theory. In particular, we shall consider various Kadec-Klee type properties, and weak and weak* normal structure. It turns out that many of these properties, though generally distinct, coincide in Orlicz sequence spaces and that all of them are intimately related to the so-called Δ 2 -condition. Some of our results extend to vector-valued Orlicz sequence spaces. For example, we prove a rather powerful theorem on the preservation of weak normal structure under the formation of substitution spaces. There is also a fixed point theorem: the Orlicz sequence space h M has the fixed point property if the complementary Orlicz function M * satisfies theΔ 2 -condition. Another one of our results implies that, under this assumption on M *, h M has weak normal structure if and only if M also satisfies the Δ 2 -condition.
Dulst et al. (Sun,) studied this question.