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

TL;DR

This paper establishes classical inequalities like Cauchy-Schwarz and Hölder within the framework of vector lattices and $f$-algebras, providing new identities, conditions for equality, and extending known results.

Contribution

It introduces a new identity for sesquilinear maps in vector lattices, reformulates inequalities for $f$-algebras, and extends Hölder and Minkowski inequalities to weighted geometric mean closed Archimedean algebras.

Findings

Proves a Cauchy-Schwarz inequality for sesquilinear maps in vector lattices.

Extends Hölder and Minkowski inequalities to weighted geometric mean closed Archimedean algebras.

Provides conditions for equality in these inequalities.

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.