Banach Journal of Mathematical Analysis

Vector lattices and f-algebras: The classical inequalities

G. Buskes and C. Schwanke

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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.

Article information

Banach J. Math. Anal. Volume 12, Number 1 (2018), 191-205.

Received: 9 February 2017
Accepted: 27 March 2017
First available in Project Euclid: 10 November 2017

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Digital Object Identifier

Primary: 46A40: Ordered topological linear spaces, vector lattices [See also 06F20, 46B40, 46B42]

vector lattice f-algebra Cauchy–Schwarz inequality Hölder inequality Minkowski inequality


Buskes, G.; Schwanke, C. Vector lattices and $f$ -algebras: The classical inequalities. Banach J. Math. Anal. 12 (2018), no. 1, 191--205. doi:10.1215/17358787-2017-0045.

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