Vector lattices and f-algebras: The classical inequalities

G. Buskes; C. Schwanke

doi:10.1215/17358787-2017-0045

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

Abstract

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 $\Phi$ -algebras, substantially improving results by K. Boulabiar and M. A. Toumi. As a consequence, a Minkowski inequality for weighted geometric mean closed Archimedean $\Phi$ -algebras is obtained.

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Citation Download Citation

G. Buskes and C. Schwanke "Vector lattices and

f

-algebras: The classical inequalities," Banach Journal of Mathematical Analysis 12(1), 191-205, (January 2018). https://doi.org/10.1215/17358787-2017-0045

Received: 9 February 2017; Accepted: 27 March 2017; Published: January 2018

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