Abstract
In this paper, we consider products K · L of unconditional bodies in ℝn. For an unconditional body M in ℝn, we introduce the notion of the M-dual Ko M of an unconditional body K: the maximal body L such that K · L ⊂ M; thus the l n 1-dual is the usual dual. We prove that if M is the unit ball of l n p , 1 ≤ p ≤ ∞, and K and L are unconditional bodies that are maximal subject to K · L ⊂ M, then K ·L = M; in other words, for any K we have \( K_M^{oo} \cdot K_M^o = M \) · K M O = M. This generalises Lozanovskii’s theorem. We also construct an example to show that equality need not hold for a general unconditional body M: \( M:\;K_M^{oo} \cdot K_M^o = M \) · K M O = M does not hold in general.
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© 1995 Birkhäuser Verlag Basel/Switzerland
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Bollobás, B., Leader, I. (1995). Products of Unconditional Bodies. In: Lindenstrauss, J., Milman, V. (eds) Geometric Aspects of Functional Analysis. Operator Theory Advances and Applications, vol 77. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9090-8_3
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DOI: https://doi.org/10.1007/978-3-0348-9090-8_3
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0348-9902-4
Online ISBN: 978-3-0348-9090-8
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