## Banach Journal of Mathematical Analysis

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

### Vector lattices and $f$-algebras: The classical inequalities

G. Buskes and C. Schwanke

#### 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.

#### Article information

**Source**

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

**Dates**

Received: 9 February 2017

Accepted: 27 March 2017

First available in Project Euclid: 10 November 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bjma/1510283129

**Digital Object Identifier**

doi:10.1215/17358787-2017-0045

**Subjects**

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

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.bjma/1510283129