A Characterization of Euclidean Spaces

The purpose of this paper is to give an elementary proof of the fact that a Banach space in which there exist projection transformations of norm one on every two-dimensional linear subspace is a euclidean space. S. Kakutani [ l ] has pointed out that a modification of a proof due to Blaschke [2] will prove this theorem. F. Bohnenblust has been able to establish this theorem for the complex case by still another method. A Banach space is a linear, normed, complete space [3, chap. 5] . A euclidean space of dimension a, where a is any cardinal number, is defined to be the Banach space of sequences xv of real numbers where v ranges over a class of cardinal number a, and ^oft is finite and equal to the square of the norm [4]. We consider only spaces having at least three linearly independent elements. P. Jordan and J. von Neumann have shown [5] that a Banach space which is euclidean in every two-dimensional linear subspace is itself a euclidean space. I t is thus sufficient to show that the "unit sphere" S for any three-dimensional linear subspace is an ellipsoid. Because of the norm properties, S is a convex body symmetric about the origin 0, and contains 0 as an interior point. Let 7 be a plane containing 0 and let Cy be the curve of intersection of 7 and the boundary 5 ' of 5. The existence for each 7 of a projection operation of norm one, whose direction of projection is that of the unit vector vy, implies that the cylinder generated by lines of direction vy tracing Cy contains 5. Our theorem is therefore an immediate consequence of the following lemma on convex bodies (which need not be symmetric about 0).