Each mathematician operating in Banaeh spaee geometry or Approximation thought is familiar with, from his personal experienee, that the majority "natural" geometrie homes may well faH to carry in a generalnormed spaee except the spaee is an internal produet spaee. To reeall the weIl recognized definitions, this implies IIx eleven = *, the place is an internal (or: scalar) product on E, Le. a functionality from ExE to the underlying (real or eomplex) box pleasurable: (i) O for x o. (ii) is linear in x. (iii) = (intherealease, thisisjust =
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4): Take 11. L = ~fJ . Lllxll 2 + (1- 'A) (1 - p,)IIyII2). 0 < x,y > < x,Y > 49 PARALLELOGRAM INEQUALITIES eH6 (The parallelogram equality is the case A = J-L = *). s. 1 + ... n = O. i iJ j i II xil1 2 = O. 3 = - 1). 4), we may assurne dimE = 2. If '" stands for ~, let 1:0 C SE be a Loewner ellipse for SE' 1:0 defines in IR2 a Euclidean norm 1·1 ~ 11·11· l:o touches SE at least at four points. 4) is violated. 6): Observe first that, by convexity of t P / 2 , the left hand side is maximizcd whcn < x ,y > > O.
E: SF and ,; > 0, take an ,;-net (vl, ... ,vj:l in SF. Choose T1, ... , TjTTj_1 ... TT1vj = IITTj_1T... TT1vjllu. Let l' = TTj: ... TT 1. 13'), 1-,; = 111'11-,; S; m~IITvjll = m~II(TTj: ... TTj + 1) T(Tj T... TT1)vjll ~ 11 Tu 11. J J Since ,; was arbitrary, 11 Tu 11 = 1. 14) IIx - Ylillz - wli + IIx - wlillY - zl/ ~ IIx - zl/ I/y - wli 'Q'x,y,z,we: E. e. the sums of the products of the opposite sides of a quadrilateral is at least the product of the diagonals). The special case z = - x, '11. 1).
Then (x - z) (y - w) (x - y) (z - w) = 1+ \ , and we take absolute values. ' .. ,E"n e: IR. 1). ; o. 0 J. : 3n ~ 2 such that cxk t k=1 > O. fJk7k cF 0, + 7kvl1 2 .... ; 2. • n) cxJ3k7k = 0 and (k and cxk,ßk,7k E: IR n ~ k=l E CXk(fJ, + 7'). : > 0, fJk1 E cxkflofJkj = 0 for i cF j and E cxkll EfJki ill k=1 k=1 i=1 U 2 .... 16): Take u2 = ... n.. 17): Immediate computation. there exist t fJki cF O. 18) For some n ~ 2. if ul • . . • U n E Sp; and fi = 1.. L)lIyIl2). 22) it sufficies to treat the case dirnE 2, A (uj.
Characterizations of Inner Product Spaces by Amir