Purely non-atomic weak $L^p$ spaces

• Autorzy: Denny H. Leung
• Języki publikacji: EN

Abstrakty

EN

Let (Ω,∑,μ) be a purely non-atomic measure space, and let 1 < p < ∞. If $L^{p,∞}(Ω,∑,μ)$ is isomorphic, as a Banach space, to $L^{p,∞}(Ω',∑',μ')$ for some purely atomic measure space (Ω',∑',μ'), then there is a measurable partition $Ω = Ω_{1} ∪ Ω_{2}$ such that $(Ω_{1}, Σ ∩ Ω_{1},μ|_{Σ ∩ Ω_{1}})$ is countably generated and σ-finite, and that μ(σ) = 0 or ∞ for every measurable $σ ⊆ Ω_{2}$. In particular, $L^{p,∞}(Ω,∑,μ)$ is isomorphic to $ℓ^{p,∞}$.

Kategorie tematyczne

• 46E30: Spaces of measurable functions ( L p -spaces, Orlicz spaces, K\"othe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
• 46B03: Isomorphic theory (including renorming) of Banach spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1997
• Tom: 122
• Numer: 1
• Strony: 55-66

Daty

wydano

1997

otrzymano

1996-01-29

poprawiono

1996-07-29

Twórcy

autor

Denny H. Leung

• Department of Mathematics, National University of Singapore, Singapore 119260

Bibliografia

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• [2] H. E. Lacey, The Isometric Theory of Classical Banach Spaces, Springer, 1974.
• [3] D. H. Leung, Isomorphism of certain weak $L^p$ spaces, Studia Math. 104 (1993), 151-160.
• [4] D. H. Leung, Isomorphic classification of atomic weak $L^p$ spaces, in: Interaction between Functional Analysis, Harmonic Analysis and Probability, N. J. Kalton, E. Saab and S. J. Montgomery-Smith (eds.), Marcel Dekker, 1996, 315-330.
• [5] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer, 1977.
• [6] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Springer, 1979.
• [7] H. P. Lotz, Weak* convergence in the dual of weak $L^p$, unpublished manuscript.
• [8] H. H. Schaefer, Banach Lattices and Positive Operators, Springer, 1974.