On geometry of Orlicz spaces
In terms of the function Φ it is established when Orlicz space LΦ does not contain l ∞ n uniformly and when it has some type or cotype.
We study some geometrical properties of Orlicz spaces. Namely using the results of  we establish when the given Orlicz space does not contain l ∞ n uniformly and when it has some type or cotype.
It is clear that every Banach space has type p,0<p⩽1.
If X is of type p, 1<p⩽2, then X* is of cotype p′=p/p−1. Đenote by l ∞ n the Rn with the maximum-norm. We shall say that a Banach space X contains l ∞ n uniformly if for each ɛ>0 and any integer n there exists an injective linear operator J: l ∞ n → X such that ‖J‖‖J−1‖<1+ɛ. X does not contain l ∞ n uniformly if and only if it has certain cotype q ().
By Δ2 we denote the family of functions Φ that satisfy the so-called Δ2 condition (i.e. Φ (2u)⩽c Φ (u) for some c>0 and every u ∈ R+). If Φ ∈ Δ2, then x ∈ LΦ if and only if ρΦ(x)<∞.
KeywordsBanach Space Measurable Function Geometrical Property Measure Space Independent Random Variable
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