Several investigations have been done on properties of derivations act¬ing on C* -algebras and their subalgebras and these include their norms, boundedness, numerical ranges, spectra and the operators inducing them. The norm property of derivations induced by normal, hyponormal, S¬universal operators among others have been considered by mathemati¬cians like Kittaneh, Mecheri, Bonyo, Agure, Okelo and others. However, little work has been done on norms of derivations implemented by norm¬attainable operators. Therefore, the objectives of the study were: To in¬vestigate tensor norms in Banach algebras, to determine upper and lower norm estimates of inner derivations implemented by norm-attainable op¬erators. The methodology involved restating known-fundamental results but omitting their proofs. The technical approaches of direct sum and po¬lar decomposition were used in investigating the norm property. Certain inequalities were also useful for example Cauchy-Bunyakovsky-Schwarz inequality, triangular inequality and Holders inequality. The results show that IIT@SII :S IITII IISII for tensor product of operators. We also showed that IIT0SII :S IITII IISII in operator spaces. For objective two we obtained that lloa,bll :S llallllbll- For objective three we have llc5a,bll 2: llallllbll- We have also given some norm inequalities for operators in norm-attainable class. The study of inner derivations is applicable in video imagery in near shore oceanographic field study, velocity spectral-digital computer deriva¬tion, embryonic stem cell line derived from human blastocysts, study on creep-fatigue evaluation procedures for high-chromium steels and the study and interpretation of the chemical characteristics of natural water amongst others.