Inclusion Theorem between the Spaces Generated by Musielak-φ Function

Abstract

Let X and G = (g i ) respectively be a Banach space and a sequence of Musielak-φ function which is the generalization of Musielak-Orlicz function. Introducing the space of all X-valued sequences generated by Musielak-φ function, denoted by E ∃ (X,G), we investigate the sufficient and necessary condition for the inclusion relation of E ∃ (X,G) ⊂ F(X) and E(X) ⊂ E ∃ (X,G) where E ∈ {ℓ ∞ , ℓ 1 , c 0 } and F ∈ {ℓ ∞ , c 0 }.