On Isosceles Orthogonality

Abstract

The ordinary length ce2+y2 of a vector in the plane may be called its Euclidean norm to distinguish it from other norms, such as lxl + lyl. Relative to other norms, there is, in general, no notion comparable to orthogonality that is nearly so useful. The following observations are related to this fact; we have found them to be interesting to students of modern algebra, and not well known to colleagues. Let V denote a vector space with real scalars, equipped with a norm jvjj. If there exists an inner product (v,w) in V such that V/(-Jv equals the norm IlvjI for all v in V, then V is said to be Euclidean. In this case (v,w) = 0 if and only if Ivl+wlI = Ilv-wlj. Thus it is possible to define orthogonality entirely in terms of the norm, and this suggests the following definition, which is due to R. C. James [3]. Vectors v and w are said to be isosceles orthogonal if 1jv+wjj = lv-wll'. Call the collection C(v) of all vectors isosceles orthogonal to a vector v the orthogonal complement of v. If V is Euclidean, isosceles orthogonality is entirely equivalent to ordinary orthogonality, and in this case C(v) is a subspace for every v. This simple observation is very useful for showing that there can exist no inner product giving rise to the norm cZ1 + yj in the plane. For let u = (2,1), v = (-I,1), and w = (1,-2); then v and w are isosceles orthogonal to u (in this norm), but v+w is not. Similar examples are easily constructed for other norms, such as max (jcej, Iyj), and this observation also provides an easy way to demonstrate that certain normed function spaces are not Hilbert spaces. THEOREM 1. A re2l normed vector space V is Euclidean if and only if C(v) is a subspace, for every element v in V. THEOREM 2. A two dimensional real normed vector space W is isometric with the Euclidean plane if and only if the orthogonal complement of every vector is a subs pace. Theorem 1 may be proved using a stronger theorem due to R. C. James [3]. His theorem depends on a theorem due to Ficken [2] theproof of which