# The dual space of L p of a vector measure

The dual space of L p of a vector measure For a vector measure ν having values in a real or complex Banach space and $${p \in}$$ [1, ∞), we consider L p (ν) and $${L_{w}^{p}(\nu)}$$ , the corresponding spaces of p-integrable and scalarly p-integrable functions. Given μ, a Rybakov measure for ν, and taking q to be the conjugate exponent of p, we construct a μ-Köthe function space E q (μ) and show it is σ-order continuous when p > 1. In this case, for the associate spaces we prove that L p (ν) ×  = E q (μ) and $${E_q(\mu)^\times = L_w^p(\nu)}$$ . It follows that $${L_p (\nu) ^{**} = L_w^p (\nu)}$$ . We also show that L 1 (ν) ×  may be equal or not to E ∞ (μ). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# The dual space of L p of a vector measure

, Volume 14 (4) – Jun 15, 2010
15 pages

/lp/springer_journal/the-dual-space-of-l-p-of-a-vector-measure-xaSNNtXu2q
Publisher
SP Birkhäuser Verlag Basel
Subject
Mathematics; Econometrics; Calculus of Variations and Optimal Control; Optimization; Potential Theory; Operator Theory; Fourier Analysis
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-010-0071-y
Publisher site
See Article on Publisher Site

### Abstract

For a vector measure ν having values in a real or complex Banach space and $${p \in}$$ [1, ∞), we consider L p (ν) and $${L_{w}^{p}(\nu)}$$ , the corresponding spaces of p-integrable and scalarly p-integrable functions. Given μ, a Rybakov measure for ν, and taking q to be the conjugate exponent of p, we construct a μ-Köthe function space E q (μ) and show it is σ-order continuous when p > 1. In this case, for the associate spaces we prove that L p (ν) ×  = E q (μ) and $${E_q(\mu)^\times = L_w^p(\nu)}$$ . It follows that $${L_p (\nu) ^{**} = L_w^p (\nu)}$$ . We also show that L 1 (ν) ×  may be equal or not to E ∞ (μ).

### Journal

PositivitySpringer Journals

Published: Jun 15, 2010

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