Cayley-Dickson Construction

Cayley-Dickson construction produces a sequence of normed algebras over real numbers. Its consequent applications result in complex numbers , quaternions, octonions, etc. In this paper we formalize the construction and prove its basic properties. The notation and terminology used here have been introduced in the following We use the following convention: u, v, x, y, z, X, Y are sets and r, s are real numbers. One can prove the following proposition (1) For all real numbers a, b, c, d holds (a + b) 2 + (c + d) 2 ≤ (√ a 2 + c 2 + √ b 2 + d 2) 2. Let X be a non trivial real normed space and let x be a non zero element of X. One can verify that x is positive. Let c be a non zero complex number. Note that c 2 is non zero.