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
-algebra,
we provide a sharpening of the triangle inequality as well as a condition for
equality.
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