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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