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arXiv · 1607.06639

Vector lattices and $f$-algebras: the classical inequalities

Gerard Buskes·Christopher Schwanke
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Abstract

We prove an identity for sesquilinear maps from the Cartesian square of a vector space to a geometric mean closed Archimedean (real or complex) vector lattice, from which the Cauchy-Schwarz inequality follows. A reformulation of this result for sesquilinear maps with a geometric mean closed semiprime Archimedean (real or complex) $f$-algebra as codomain is also given. In addition, a sufficient and necessary condition for equality is presented. We also prove the H\"older inequality for weighted geometric mean closed Archimedean (real or complex) $\Phi$-algebras, improving results by Boulabiar and Toumi. As a consequence, the Minkowski inequality for weighted geometric mean closed Archimedean (real or complex) $\Phi$-algebras is obtained.

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BibTeXRIS

Gerard Buskes, Christopher Schwanke. 2016-07-22. Vector lattices and $f$-algebras: the classical inequalities. https://doi.org/10.1215/17358787-2017-0045

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