2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/126032We introduce two kinds of operator-valued norms. One of them is an $L(H)$-valued norm. The other one is an $L(C(K))$-valued norm. We characterize the completeness with respect to a bounded $L(H)$-valued norm. Furthermore, for a given Banach space $\textbf{B}$, we provide an $L(C(K))$-valued norm on $\textbf{B}$. and we introduce an $L(C(K))$-valued norm on a Banach space satisfying special properties.8 pagesFunctional AnalysisOperator Algebras47A30; 46C05; 46B99; 46J10text