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January 2018 Vector lattices and f-algebras: The classical inequalities
G. Buskes, C. Schwanke
Banach J. Math. Anal. 12(1): 191-205 (January 2018). DOI: 10.1215/17358787-2017-0045


We present some of the classical inequalities in analysis in the context of Archimedean (real or complex) vector lattices and f-algebras. In particular, we prove an identity for sesquilinear maps from the Cartesian square of a vector space to a geometric mean closed Archimedean vector lattice, from which a Cauchy–Schwarz inequality follows. A reformulation of this result for sesquilinear maps with a geometric mean closed semiprime Archimedean f-algebra as codomain is also given. In addition, a sufficient and necessary condition for equality is presented. We also prove a Hölder inequality for weighted geometric mean closed Archimedean Φ-algebras, substantially improving results by K. Boulabiar and M. A. Toumi. As a consequence, a Minkowski inequality for weighted geometric mean closed Archimedean Φ-algebras is obtained.


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G. Buskes. C. Schwanke. "Vector lattices and f-algebras: The classical inequalities." Banach J. Math. Anal. 12 (1) 191 - 205, January 2018.


Received: 9 February 2017; Accepted: 27 March 2017; Published: January 2018
First available in Project Euclid: 10 November 2017

zbMATH: 06841271
MathSciNet: MR3745580
Digital Object Identifier: 10.1215/17358787-2017-0045

Primary: 46A40

Keywords: Cauchy–Schwarz inequality , F-algebra , Hölder inequality , Minkowski inequality , ‎vector lattice‎‎

Rights: Copyright © 2018 Tusi Mathematical Research Group

Vol.12 • No. 1 • January 2018
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