A characterization of continuous module homomorphisms on random semi-normed modules and its applications
The authors prove the following main theorem: Suppose \((S,\{\chi^d\}_{d\in D})\) is a random semi-normed module with base \((\Omega,{\mathcal A},\mu)\) over \(K\), and \(f: S\to L(\mu, K)\) is a linear mapping. Then the following statements are equivalent: (1) \(f\) is a continuous module homomorphism; (2) there exist a countable partition \(\{A_i\mid i\in\mathbb{N}\}\) of \(\Omega\) to \({\mathcal A}\), a sequence \(\{C_i\mid i\in\mathbb{N}\}\) in \(L^+(\mu)\) with the \(\mu\)-essential support of \(C_i\) contained in \(A_i\) for each \(i\in\mathbb{N}\), and a countable subfamily \(\{H_i\mid i\in\mathbb{N}\}\) of \({\mathcal F}(D)\) such that \(|f(p)|\leq \sum^\infty_{i=1} C_i\cdot X^{H_i}_p\), \(\forall p\in S\), where \({\mathcal F}(D)\) stands for the collection of all finite subsets of \(D\), and \(\chi^{H_i}: S\to L^+(\mu)\) is defined by \(X^{H_i}_p= \chi^{H_i}(p)= \sum_{d\in H_i} X^d_p\), \(\forall i\geq 1\) and \(p\in S\), or by \(X^{H_i}_p= \sum_{d\in H_i} X^d_p\), \(\forall i\geq 1\) and \(p\in S\); (3) there exists a continuous random seminorm \(F\) on \(S\) satisfying \(F(\xi\cdot p)= \xi\cdot F(p)\), \(\forall\xi\in L^+(\mu)\) and \(p\in S\), such that \(|f(p)|\leq F(p)\), \(\forall p\in S\).
scientific article; zbMATH DE number 1921190