On continuous preservation of norms and areas

Summary

We give various properties, examples and equivalent conditions for mapsT of then-dimensional euclidean space into itself (n ⩾ 2) satisfying the generalised orthogonality equation|Tx ⋅ Ty| = |x ⋅ y| for allx, y inR ⁿ, where ⋅ stands for the usual dot product, and we prove that the only continuous maps verifying this condition are the orthogonal linear transformations.