If A is a normed algebra, a net {e_i} in A is called an approximate identity for A if sup_i left bracketing bar e_i right bracketing bar <∞ and if for each a in A, e_i a->a and a e_i->a. Though this definition makes sense framed with respect to any normed algebra A, it is usually delivered in the specific cases that A is either a Banach algebra or a local Banach algebra. Note that if only e_i a->a (a e_i->a, respectively), then {e_i} is said to be a left (respectively, right) approximate identity.

algebra | Banach algebra | local Banach algebra | net | normed space | real normed algebra