Each mathematician operating in Banaeh spaee geometry or Approximation conception is familiar with, from his personal experienee, that the majority "natural" geometrie houses could faH to carry in a generalnormed spaee until the spaee is an internal produet spaee. To reeall the weIl identified definitions, this suggests IIx eleven = *, the place is an internal (or: scalar) product on E, Le. a functionality from ExE to the underlying (real or eomplex) box gratifying: (i) O for x o. (ii) is linear in x. (iii) = (intherealease, thisisjust =
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11) Fi 1 is dissipative "V' convex AcE. 11') F[-;,1y ] is dissipative "V'x,y E E. 1): Let p 11 q, x == p + q, Y == P - q. Then fx ,y J c F[z,tI] F[z,tI]P, hence p~(2q ,p) ~ and p~(2q, - p) ~ 0, so that q lp. 1): Same (getting p lqJ. , YEFAx~
E. vol t;" > vol~, unless cl = 1 for all i. If span(S n t;) is not n-dimensional, we can stretch ~ in the orthogonal direction and get a larger volume. 1, and let 11,11, I· I be the corresponding norms. rn IIx 11 V'x E E. 2: Proof: IIxll ~ Ix I since ~o eS. For the second inequality, we may assume (by making an affine transformation) that t;o is the Euclidean unit sphere and that x = de l' For any 2-dimensional subspace F==span(e1'Y) of E and tE(O,l], let Tt(ael+ßy)==ate1+ßY. 1). Therefore, Eil' 11 contains the ellipsoid t2~r + f: k=2 ~l:s; Pr.
Iv -*l1il. ,~. 112u -vII = 112v -ull· E SE' Then 0 hc "'-:w,itten as: x + Y + z = o. 20) x + Y + z + w = J. IIxll = lIylL IIzll = IIwll => IIx - zll = lIy - wll. 20): ~ IIx - z 11 2 = lIy - w1l 2 . 21) There are a maximal subspace H of E and a unit vector such that span(u,x) is Euclidean V'x E E. 1): Let H' be a maximal subspaee with u lH'. Then IIsu+txll 2 = s2 + t 211xll 2 V'x EH'. li x,Y EH' and x-o:u,Y-ßu EH then IIx +y 11 2 + (0:+ß)2 + IIx-ylI 2 + (0:-ß)2 = lI(x-o:u)+(y-ßu)1I 2 + lI(x-o:u)-(y-ßu)1I 2 = 211x-o:u1I 2 + 2l1y-ßulI = 211xll 2 + 20:2 + 211yII 2 + 2ß2 henee IIx +y 11 2 + IIx-YIl2 = 211xll 2 + 211Y1l2, so that we may assume H' = H.
Characterizations of Inner Product Spaces by Amir