On isometries of product sets
Paul J. Kelly · Bulletin of the American Mathematical Society · 1948
Introduction.If A is a metric set, and A 2 its cartesian product with itself, it is possible, in terms of the metric A, to metrize A 2 in many different ways.Thus, the problem of determining when A 2 isometric to B 2 implies A isometric to B can be attacked by placing limitations on the sets A and B or on the product metrics.The present paper proves that the isometry is implied when A and B are bounded and linear and the metric in A 2 and B 2 is determined by the use of a modified Minkowski gauge.Also stated, without proof, are certain conditions under which the isometry of two product sets implies the isometry of the factors.The proofs for these results, given in the author's doctoral thesis, 1 are long and differ only in detail from the one demonstrated.