A geometric theorem and its application to biorthogonal systems

1. The geometric theorem. Let En denote an w-dimensional Euclidean space. A set of points in En may constitute an En-\\ if so, it will be called a plane in En. We shall sometimes denote planes by pu p2i • • • • Also, if xi, • • • , Xn-i are linearly independent vectors emanating from a common point O in En, the plane through O and containing these vectors will be denoted by E{x\, • • • , #n-i). A plane divides En into two closed half-spaces each having the plane as boundary, but otherwise having no points in common. If 5 is a point set in Eni a plane p is called a supporting plane of S if (1) S lies in one of the closed half-spaces determined by p, and (2) the distance between S and p is zero.