Inner Products in Normed Linear Spaces

Let T be any normed linear space [l, p. S3]. Then an inner product is defined in T if to each pair of elements x and y there is associated a real number (x, y) in such a way that (#, y) » (y, x), \\x\\ = (#, #), (x, y+z) = (#,y) + (x, 2), and (/#,y) = /(#, y) for all real numbers /and elements x and y. An inner product can be defined in T if and only if any two-dimensional subspace is equivalent to Cartesian space [5]. A complete separable normed linear space which has an inner product and is not finite-dimensional is equivalent to (real) Hubert space, while every finite-dimensional subspace is equivalent to Euclidean space of that dimension. Any complete normed linear space T which has an inner product is characterized by its (finite or transfinite) cardinal "dimension-number" n. It is equivalent to the space of all sets x = (xi, #2, • • • ) of n real numbers satisfying ]T)< a? < + 00, where \\x\\ — (X^*?)' [7, Theorem 32]. Various necessary and sufficient conditions for the existence of an inner product in normed linear spaces of two or more dimensions are known. Two such conditions are that | |x+y| | +| |^-y | | 2 = 2[||x||+j|y||] for all x and y, and that limn^oo||^+wy||—||«x+y|| = 0 whenever \\x\\ =||y|| ([5] and [4, Theorem 6.3]). A characterization of inner product spaces of three or more dimensions is that there exist a projection of unit norm on each twodimensional subspace [6, Theorem 3]. Other characterizations valid for three or more dimensions will be given here, expressed by means of orthogonality, hyperplanes, and linear functionals. A hyperplane of a normed linear space is any closed maximal linear subset M, or any translation x+M of M. A hyperplane is a supporting hyperplane of a convex body S if its distance from S is zero and it does not contain an interior point of 5; it is tangent to 5 at x if it is the only supporting hyperplane of S containing x [8, pp. 70-74]. It will be said that an element #0 of T is orthogonal to y (xoJ~y) if and only if ||#o+&y|| è||#o|| for all k, which is equivalent to requiring the existence of a nonzero linear functional ƒ such that ƒ (xo) — Il/Il ll*o|| * f(y) =0, or that xo+y belong to a supporting hyperplane of the sphere