Abstract

We give a simple proof of a characterization of euclidean space due to Aronszajn and derive a well-known characterization due to Jordan & von Neumann as a corollary. A norm | | | | on a vector space V is euclidean if there is an inner product 〈 , 〉 on V such that ||v| | = √ 〈v,v〉. Characterizations of euclidean normed spaces abound. Amir [1] surveys some 350 characterizations, starting with a wellknown classic of Jordan & von Neumann [3]: a norm is euclidean iff it satisfies the parallelogram identity: ||v + w| | = 2||v| | 2 + 2||w| | 2 − ||v − w| | 2 Aronszajn proved that the algebraic details of this identity are mostly irrelevant: if the norms of two sides and one diagonal of a parallelogram determine the norm of the other diagonal then the norm is euclidean. Formally, Aronszajn’s criterion is the following property, as illustrated in figure 1(a). ∀v1 w1 v2 w2 · ||v1| | = ||v2| | ∧ ||w1| | = ||w2| | ∧ ||v1 − w1| | = ||v2 − w2|| ⇒ ||v1 + w1| | = ||v2 + w2||. Aronszajn’s announcement of this characterization [2] does not give a proof. Amir’s proof forms part of a long chain of interrelated results. In this note we give a short, self-contained proof of the theorem and derive the Jordan-von Neumann theorem as a corollary. We begin with a lemma showing that the Aronszajn criterion ensures a useful supply of isometries. Figure 1(b) illustrates the parallelograms that feature in the proof. This note was inspired by joint work with Robert M. Solovay and John Harrison on decidability for logical theories of normed spaces. I am grateful to Bob and John for ther comments. 1 ap + (b+1)q ap + bq 0 v2 w1 v1+ w1 v 1 v 1 w2 v2 v2 − w1 w2 w2

Citations