Each mathematician operating in Banaeh spaee geometry or Approximation thought is familiar with, from his personal experienee, that almost all "natural" geometrie homes may perhaps faH to carry in a generalnormed spaee until the spaee is an internal produet spaee. To reeall the weIl recognized 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 enjoyable: (i) O for x o. (ii) is linear in x. (iii) = (intherealease, thisisjust =
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Extra resources for Characterizations of Inner Product Spaces
If u,v E SE then u lx and v lx for some 0 #- x E span(u,v) (v iff [u ,v] (v) C SE or [u, -v] C SE' E i:s :sLricLly convex irr orLhogorwliLy is unique from Lhe lefl in every 2-dimensional subspace iff for every maximal subspace H there is at most one I-dimensional subspace L with L lH. , then x ly ~> < x ,y > = O. 8) are: ('1. l) x # y (4 ;~) (4 :3) > x ly. (u + v) l(U - v) rOh] Vu,v E ~. (x + fucll) lIyl! y 1 (x - fucll) lIylI y [Day2], [Oh] Vx,y #- O. 4) x Ly ~ xII y. 4), 112w 2w - U E (w,v), and this is impossible.
Ai XjIl2 + i=l ~ vtl 2 .... 4 ~. Vu,v Es,; 3 and all' .. an I E lS:i
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. Now repeat the same eomputation for any x,Y EE. s. 4): Sinee every 2-dimensional subspaee of E is eontained in the span of y and finitely many of the x a ' we may assume A finite.