A characterization of inner product spaces

Abstract

In this paper we define a generalized inner product on an arbitrary normed linear space and use this generalized inner product to characterize inner product spaces in the class of all normed linear spaces. We give a sharp statement of a generalized Riesz representation theorem for bounded linear functionals. This theorem should be useful in generalizing the notions of gradient methods and reproducing kernel spaces. 1. Main result. We first prove the results stated in [12]. Let X be a normed linear space andf a functional defined on X. Recall that by the first right-hand Gateaux derivative of f at x in the direction h we mean (1) f$+(x)(h) = lim t-'[f(x + th) -f(x)], t--I+0 with the obvious definition forfL'(x)(h). Whenf+(x)(h)=fJ'(x)(h) we say thatf is Gateaux differentiable at x in the direction h. Letf(x)=iI1xII2 and (x, y) =f+(x)(y). PROPOSITION 1. Every normed linear space is a generalized inner product space in the sense that (a) (x, y) is well defined. (b) 1lX 1=(X, X)1/2. (c) I (x, y)II lxii Ilyll (Cauchy-Schwarz-Buniakovsky inequality). (d) If X is an inner product space with inner product [x, y], then (x, y)= [x, y]. PROOF. For x, y E X and t>O let R(t, y)=(t-I(Ix+tylI-IIxll). From Mazur [9, p. 75] we have that (2) (i) R(t, y) is nondecreasing in t, and ( ) ~~~(ii) -lylYl < R(t, y)-< llYllIt follows that T(y)=limt,o R(t, y) exists. Also (3) (x,y)=liml(lIx + tyll + llxll)lim t1(Ilx + tyll lxll) = lixll T(y). t g+0 t--+O Received by the editors April 2, 1973. AMS (MOS) subject classifications (1970). Primary 46B99, 46CI0.