バナッハ環
Any Banach algebra formula_2 (whether it has an identity element or not) can be embedded isometrically into a unital Banach algebra formula_3 so as to form a closed ideal of formula_3. Often one assumes "a priori" that the algebra under consideration is unital: for one can develop much of the theory by considering formula_3 and then applying the outcome in the original algebra. However, this is not the case all the time. For example, one cannot define all the trigonometric functions in a Banach algebra without identity.