A formula for angles between subspaces of inner product spaces.

Abstract. We present an explicit formula for angles between two subspaces ofinner product spaces. Our formula serves as a correction for, as well as an extensionof, the formula proposed by Risteski and Trenˇcevski [13]. As a consequence of ourformula, a generalized Cauchy-Schwarz inequality is obtained.MSC 2000: 15A03, 51N20, 15A45, 15A21, 46B20Keywords: Angles between subspaces, canonical angles, generalized Cauchy-Schwarz inequality1. IntroductionThe notion of angles between two subspaces of the Euclidean space R d has been studiedby many researchers since the 1950’s or even earlier (see [3]). In statistics, canonical (orprincipal) angles are studied as measures of dependency of one set of random variables onanother (see [1]). Some recent works on angles between subspaces and related topics canbe found in, for example, [4, 8, 12, 13, 14]. Particularly, in [13], Risteski and Trenˇcevskiintroduced a more geometrical definition of angles between two subspaces of R d and explainedits connection with canonical angles. Their definition of the angle, however, is based on ageneralized Cauchy-Schwarz inequality which we found incorrect. The purpose of this noteis to fix their definition and at the same time extend the ambient space to any real innerproduct space.Let (X,h·,·i) be a real inner product space, which will be our ambient space throughoutthis note. Given two nonzero, finite-dimensional, subspaces U and V of X with dim(U) ≤

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