#### 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
Primary: 46A40