Vector lattices and $f$-algebras: the classical inequalities
Functional Analysis
2018-02-21 v2
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) -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) -algebras, improving results by Boulabiar and Toumi. As a consequence, the Minkowski inequality for weighted geometric mean closed Archimedean (real or complex) -algebras is obtained.
Keywords
Cite
@article{arxiv.1607.06639,
title = {Vector lattices and $f$-algebras: the classical inequalities},
author = {Gerard Buskes and Christopher Schwanke},
journal= {arXiv preprint arXiv:1607.06639},
year = {2018}
}