A Pythagorean inequality

Abstract

Abstract. Let {vx, v2 , v$ , ... } be a sequence of elements of a Hilbert space, and suppose that (one or both of) the inequalities d2J2aj < \[52a¡v¡\\2 < D2 £] aj hold for every finite sequence of scalars {a,} . If an element v0 is adjoined to {v¡} , then the resulting set satisfies (one or both of) d2JA,af < Eai'vill2 ^ ^o ¿A,a2 » where, denoting the norm of vo by r and its distance from the closed linear span of the v¡ by ô , d2 = d2 and + Ur2-d2-yV2 + d2)2 -4d262\ D2 = D2 + X-(r2 -D2 + y/(r2 + D2)2 -^D2S2\ .Both bounds are best possible. If Vq is in the span of the original set, the expressions above simplify to do = 0 and D2, = D2 + r2 . If the original set is a single unit vector vx , so d = D = 1 , and if ^o-Lvi is a unit vector so ô = X , then the above is (a2 + b2) < \\avo + bvx\\2 < (a2 + b2), the Pythagorean Theorem.Several consequences are deduced. If v¡ are unit vectors, 5Z aJ = 1 , and (5, is the distance from v¡ to the span of its predecessors (so that the volume of the parallelotope spanned by the v¡ is V" = 6xô2 ■ ■ ■ ôn), the above result is used to show that ||¿"=oa>v<ll t v*/2n/1 ■