On some properties of Musielak-Orlicz sequencespaces

摘要

We consider a nontrivial vector space X and a semimodular M : X←[0, ∞]with property: (V_x∈X)(3α> 0)M(ax) < ∞(in other words,Mis normal (i.e.(Vx E X {O}) > 0) M(ax) > 0) pregenfunction). The function M generates inX a metric d withd(x, y) := inf{a > 0 : M(a~(-1)(x -y)) 0) (3m ∈N) Σ_n>mM(α(n)) <∞and prove that(G_M, p)is separable ifand only if (X, d) is the same. Several examples are considered.