Kekonvergenan Norma Hasil Bagi dan Kekontinuan Seminorma - Rosmaya, Dwi Ajeng; Maharani, Dian (Mathematics Department, Maulana Malik Ibrahim State Islamic University of Malang, 2024)
Consider a vector space over the field of real or complex numbers. A function is said to be a seminorm on a vector space if it satisfies the conditions of a seminorm. A seminorm is a generalization of a norm. The quotient norm is defined on the quotient space formed from a normed space with a closed subspace, where the convergence property of the quotient norm is examined through the convergence of sequences in the normed space. This study aims to explore the convergence properties of the quotient norm and, regarding seminorms, to examine properties of convexity, absorption, and balance, the continuity of seminorms, as well as the relationship between seminorms and norms. The convergence property of the quotient norm indicates that a sequence will converge in the quotient space if and only if there exists a sequence in the closed subspace in the normed space such that it converges in the normed space. Furthermore, a seminorm on a vector space satisfies properties of convexity, absorption, and balance; a seminorm on a normed space will be continuous in that normed space and is formed by a linear mapping.