Ulam Stability in Real Inner-Product Spaces

Abstract

Roughly speaking an equation is called Ulam stable if near each approximate solution of the equation there exists an exact solution. In this paper we prove that Cauchy-Schwarz equation, Ortogonality equation and Gram equation are Ulam stable.

This paper is concerned with the Ulam stability of some classical equations arising in thecontext of inner-product spaces. For the general notion of Ulam stability see, e.q., [1]. Roughlyspeaking an equation is called Ulam stable if near every approximate solution there exists anexact solution; the precise meaning in each case presented in this paper is described in threetheorems. Related results can be found in [2, 3, 4]. See also [5] for some inequalities in innerproduct spaces.

Keywords

• Ulam stability
• Ortogonality equation
• Gram equation

References

1. J. Brzde¸k, D. Popa, I. Ra¸sa and Xu. B: Ulam stability of operators, Academic Press, London(2018).
2. D. S. Marinescu, M. Monea and C. Mortici: Some characterizations of inner product spaces via some geometrical inequalities, Appl. Anal. Discrete Math., 11 (2017), 424-433.
3. N. Minculete: Considerations about the several inequalities in an inner product space, Math. Inequalities, 1 (2018), 155– 161.
4. S. M. S. Nabavi: On mappings which approximately preserve angles, Aequationes Math. 92 (2018), 1079–1090.
5. D. Popa and I. Ra¸sa: Inequalities involving the inner product. JIPAM, 8 (3) (2007), Article 86.