From: jhnieto@my-dejanews.com Subject: Re: Which normed vector spaces are inner product spaces? Date: Tue, 02 Feb 1999 13:33:08 GMT Newsgroups: sci.math To: chanita@flash.net Keywords: Parallelogram law implies an inner product exists In article <36B54A5B.E6AD0AB4@flash.net>, Chanita Chantaplin wrote: > > I have been asked to show that an inner product can be defined on normed > > spaces which obey the parallelagram law: > > > > || f + g ||^2 + || f - g ||^2 = 2 || f ||^2 + 2 || g ||^2 > > > > It seems to me that the way you would do this is to define a function by > > the polerization formula (assume for simplicity that the scalar field is > > the reals): > > > > = ( || f +g ||^2 - || f-g ||^2 ) / 4 > > > > and then show that this function satisfies all axioms of an inner > > product. The axioms I am having trouble with are > > > > < kf, g > = k > > < f+g, h > = < f,h > + < g,h > > > First add and subtract ||f-g+h||^2 to 4 and apply the polar identity to obtain: 4 = ||f+g+h||^2 - ||f+g-h||^2 = ||f+g+h||^2 - ||f-g+h||^2 + ||f-g+h||^2 - ||f+g-h||^2 = 2(||f+h||^2 + ||g||^2) - 2(||f||^2 + ||g-h||^2) (1) Now permute f and g: 4 = 2(||g+h||^2 + ||f||^2) - 2(||g||^2 + ||f-h||^2) (2) Sum (1) and (2) and divide by 8 to obtain = (1/4)(||f+h||^2 - ||f-h||^2) + (1/4)(||g+h||^2 - ||g-h||^2) = + From this identity it follows easily (by induction) that = n for all positive integers n. From the definition of we see that <0f,h> = 0 = 0 and <-f,h> = -, hence = n holds for all integers. if m and n are integers, n>0, then n<(m/n)f,h> = = = m hence <(m/n)f,h> = (m/n) Finally observe (i.e. prove!) that depends continuously of f, hence if q_n is a sequence of reals that converges to a real number r we have = lim = lim q_n = r Greetings, Jose H. Nieto -----------== Posted via Deja News, The Discussion Network ==---------- http://www.dejanews.com/ Search, Read, Discuss, or Start Your Own