In this paper we carefully investigated some relationships between Birkhoff orthogonality duality map, and isosceles orthogonality, pythagorean orthogonality, and Roberts orthogonality, some characteristics of inner product spaces are also given.

This paper deduced the Pythagorean triple group formula and the double angle formula for trigonometric function via two different expressions(algebraic form and trigonometric form) of complex numbers and through the process of finding the modulus by using the binomial theorem.

Space pythagorean theorem and space pythagorean number;

The enlightenment of the evolvement of Pythagorean theorem to the modern mathematics teaching;

Space pythagorean theorem and space pythagorean number;

Let (E,S,Ω,f)be a random inner product space, the scharwz inequality, Riesz theorem, right angles theorem and some other results in (E,S,Ω,f) are proved.

From view of multi-culture, the right angles theorem was all inheritance of common mankind, it was for the tree indivisibility of world mathematics and international community all very pay attention to its value of social culture, almost whole world high school mathematics course was all introductive contents.

This article gives out a method which used basic solutions of Pythagorean equation to solve this question completely.