Vector semi-inner products

TL;DR

This paper introduces vector semi-inner products and vector seminorms, generalizing classical geometric laws like the Pythagorean theorem and parallelogram law within a new algebraic framework.

Contribution

It formalizes vector semi-inner products and develops a new class of vector seminorms based on these, extending classical geometric results to more general algebraic structures.

Findings

Generalized Pythagorean theorem for vector seminorms

Extended parallelogram law in vector lattice contexts

Sharpened triangle inequality with equality conditions

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 that have a geometric mean closed vector lattice for codomain. 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.