University of Białystok; Summary. This article contains the Hahn Banach theorem in the vector space over
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, , , and  provide the notation and terminology for this paper. The following propositions are true: 1. PRELIMINARIES (1) For every element z of C holds ||z| | = |z|. (2) For all real numbers x1, y1, x2, y2 holds (x1 + y1i) ·(x2 + y2i) = (x1 · x2 − y1 · y2)+(x1 · y2 + x2 · y1)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. 2. SOME FACTS ON THE FIELD OF COMPLEX NUMBERS Let x, y be real numbers. The functor x+yiCF yielding an element of CF is defined as follows: (Def. 1) x+yiCF = x+yi. The element iCF of CF is defined as follows: (Def. 2) iCF = i. We now state several propositions: (6) iCF = 0+1i and iCF = 0+1iCF.