Abstract
In 1966, Gleason published some interesting results about quadratic mappings. He considered a mapping Q : V → W such that Q(u + v) + Q(u − v) = 2Q(u) + 2Q(v), and he proved that it was \({\mathbb{Z}}\) -quadratic, provided that the multiplication by 2 in W was injective. Then he answered this question negatively: if V and W are vector spaces over a field K, and if Q is \({\mathbb{Z}}\)-quadratic and K-homogeneous of order 2, is it always true that Q is K-quadratic? These considerations and other related topics are here revisited, without the systematic hypothesis about the multiplication by 2.
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