Isomorphic Classification of Atomic Weak L Spaces
نویسنده
چکیده
Let (Ω,Σ, μ) be a measure space and let 1 < p < ∞. The weak L space L(Ω,Σ, μ) consists of all measurable functions f such that ‖f‖ = sup t>0 t 1 p f∗(t) < ∞, where f∗ is the decreasing rearrangement of |f |. It is a Banach space under a norm which is equivalent to the expression above. In this paper, we pursue the problem of classifying weak L spaces isomorphically when (Ω,Σ, μ) is purely atomic. It is also shown that if (Ω,Σ, μ) is a countably generated σ-finite measure space, then L(Ω,Σ, μ) (if infinite dimensional) must be isomorphic to either l∞ or l.
منابع مشابه
Purely Non-atomic Weak L P Spaces
Let (Ω,Σ, μ) be a purely non-atomic measure space, and let 1 < p < ∞. If L(Ω,Σ, μ) is isomorphic, as a Banach space, to L(Ω,Σ, μ) for some purely atomic measure space (Ω,Σ, μ), then there is a measurable partition Ω = Ω1 ∪Ω2 such that (Ω1,Σ ∩ Ω1, μ|Σ∩Ω1) is countably generated and σ-finite, and that μ(σ) = 0 or ∞ for every measurable σ ⊆ Ω2. In particular, L(Ω,Σ, μ) is isomorphic to l.
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