Vol. 15, No. 2, 2022

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Vector semi-inner products

Kyle Rose, Christopher Schwanke and Zachary Ward

Vol. 15 (2022), No. 2, 289–297
Abstract

We formalize the notion of vector semi-inner products and introduce a class of vector seminorms which are built from these maps. The classical Pythagorean theorem and parallelogram law are then generalized to vector seminorms whose codomain is a geometric mean closed vector lattice. In the special case that this codomain is a square root closed, semiprime f-algebra, we provide a sharpening of the triangle inequality as well as a condition for equality.

Keywords
vector lattice, semi-inner product, Pythagorean theorem, parallelogram law
Mathematical Subject Classification
Primary: 46A40
Milestones
Received: 29 April 2021
Revised: 21 September 2021
Accepted: 20 October 2021
Published: 29 July 2022

Communicated by Mohammad Sal Moslehian
Authors
Kyle Rose
Department of Mathematics
Lyon College
Batesville, AR
United States
Christopher Schwanke
Department of Mathematics and Applied Mathematics
Hatfield Campus
University of Pretoria
Pretoria
South Africa
Zachary Ward
Department of Mathematics
Lyon College
Batesville, AR
United States