Aequationes Math. 64 (2002) 62–69 0001-9054/02/010062-8 c  Birkh¨ auser Verlag, Basel, 2002 Aequationes Mathematicae On the Hyers–Ulam stability problem for quadratic multi-dimensional mappings John Michael Rassias Summary. In 1940 S. M. Ulam proposed the well-known Ulam stability problem. In 1941 D. H. Hyers solved this problem for linear mappings. According to P. M. Gruber (1978) this kind of stability problems is of particular interest in probability theory and in the case of functional equations of different types. In 1982–1999 we solved the above Ulam problem for different map- pings. In this paper we solve the Hyers–Ulam stability problem for quadratic multi-dimensional mappings. Mathematics Subject Classification (2000). 39B. Keywords. General stability problem, Hyers–Ulam stability, square of the quadratic weighted mean, quadratic mapping. 1. Introduction In 1940 and in 1968 S. M. Ulam [16] proposed the general Ulam stability problem: “When is it true that by slightly changing the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or approximately true?” In 1941 D. H. Hyers [4] solved this problem for linear mappings. According to P. M. Gruber [3] (1978) this kind of stability problems is of particular interest in probability theory and in the case of functional equations of different types. In 1995 G. L. Forti [2] published a survey paper which contained more information than any other publication relevant to the theory of functional equations. In 1982–1999 we ([5]–[15]) solved the above Ulam problem for different mappings. In particular, in 1996 we [13] solved the Hyers–Ulam stability problem for quadratic mappings Q : X → Y satisfying the quadratic functional equation Q(a 1 x 1 + a 2 x 2 )+ Q(a 2 x 1 − a 1 x 2 )=(a 2 1 + a 2 2 )[Q(x 1 )+ Q(x 2 )] for every x 1 ,x 2 ∈ X, and fixed reals a 1 ,a 2 = 0, where X and Y are real linear spaces. In this paper we solve the Hyers–Ulam stability problem for quadratic