  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.
