The Hahn Banach Theorem in the Vector Space over the Field of Complex Numbers

This article contains the Hahn Banach theorem in the vector space over the field of complex numbers. provide the terminology and notation for this paper. The following propositions are true: (1) For every element z of C holds ||z|| = |z|. (2) For all elements x 1 , y 1 , x 2 , y 2 of R holds (x 1 + y 1 i) · (x 2 + y 2 i) = (x 1 · x 2 − y 1 · y 2) + (x 1 · y 2 + x 2 · y 1)i. (3) For every real number r holds (r + 0i) · i = 0 + ri. (4) For every real number r holds |r + 0i| = |r|. (5) For every element z of C such that |z| = 0 holds |z| + 0i = z * |z|+0i · z.