Orthogonality in normed linear spaces
Let B be a real normed l inear space. We will say t ha t B is Eucl idean if the re is a symmet r i c bi l inear funct ional (u, v) (called the inner p roduc t of u and v) defined for u, v e B , such t h a t ( u , u ) = l l u l l 2 for every u e B . In a Euc l idean space we have the cus tomary def ini t ion of or thogonal i ty , viz. an c lement u is o r thogona l to an e lement v (in n o t a t i o n u • if and only if (u, v)=O. I t is an immed ia t e consequence of this defini t ion t ha t o r thogona l i ty has the following propert ies : (a) I f u • then v_l_u (Symmetry) ; (b) If u• then ~u,• for all real A (Homogenei ty) ; (c) If u • and v_l_w, then ( u + v ) • (Addi t iv i ty) ; (d) Fo r every pa i r u, v E B there is a number a such tha t u,i , (au+v). I f a~, by, c~, v = 1, 2, . . . , m, are real numbers sat isfying