Quadratic mappings associated with inner product spaces
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초록
In [7], Th.M. Rassias proved that the norm defined over a real vector space V is induced by an inner product if and only if for a fixed integer n ≥ 2[수식]holds for all x_1, …, x_n ∈ V. Let V, W be real vector spaces. It is shown that if an even mapping f : V → W satisfies [수식]for all x_1, …, x(_2n) ∈ V, then the even mapping f : V → W is quadratic. Furthermore, we prove the generalized Hyers-Ulam stability of the quadratic functional equation (0.1) in Banach spaces.
Abstract
In [7], Th.M. Rassias proved that the norm defined over a real vector space V is induced by an inner product if and only if for a fixed integer n ≥ 2[수식]holds for all x_1, …, x_n ∈ V. Let V, W be real vector spaces. It is shown that if an even mapping f : V → W satisfies [수식]for all x_1, …, x(_2n) ∈ V, then the even mapping f : V → W is quadratic. Furthermore, we prove the generalized Hyers-Ulam stability of the quadratic functional equation (0.1) in Banach spaces.
참고문헌 (15)
- T. Aoki, "On the stability of the linear transformation in Banach spaces", J. Math. Soc. Japan, 2, pp.64~66 (1950)
- P.W. Cholewa, "Remarks on the stability of functional equations", Aequationes Math, 27, pp.76~86 (1984)
- S. Czerwik, "On the stability of the quadratic mapping in normed spaces", Abh. Math. Sem. Univ. Hamburg, 62, pp.59~64 (1992)
- P. Gavruta, "A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings", J. Math. Anal. Appl, 184, pp.431~436 (1994)
- D.H. Hyers, "On the stability of the linear functional equation", Proc. Natl. Acad. Sci. USA, 27, pp.222~224 (1941)
- Th.M. Rassias, "On the stability of the linear mapping in Banach spaces", Proc. Amer. Math. Soc, 72, pp.297~300 (1978)
- Th.M. Rassias, "New characterizations of inner product spaces", Bull. Sci. Math, 108, pp.95~99 (1984)
- Th.M. Rassias, "On the stability of the quadratic functional equation and its applications", Studia Univ. Babes-Bolyai, XLIII, pp.89~124 (1998)
- Th.M. Rassias, "The problem of S.M. Ulam for approximately multiplicative mappings", J. Math. Anal. Appl, 246, pp.352~378 (2000)
- Th.M. Rassias, "On the stability of functional equations in Banach spaces", J. Math. Anal. Appl, 251, pp.264~284 (2000)
- Th.M. Rassias, "On the stability of functional equations and a problem of Ulam", Acta Appl. Math, 62, pp.23~130 (2000)
- Th.M. Rassias, "On the Hyers-Ulam stability of linear mappings", J. Math. Anal. Appl, 173, pp.325~338 (1993)
- Th.M. Rassias, "Variational problem of some quadratic functionals in complex analysis", J. Math. Anal. Appl, 228, pp.234~253 (1998)
- F. Skof, "Proprieta locali e approssimazione di operatori", Rend. Sem. Mat. Fis. Milano, 53, pp.113~129 (1983)
- S.M. Ulam, "Problems in Modern Mathematics" (1960), Wiley, New York