Approximation of the identity

In mathematics, particularly in functional analysis and ring theory, an approximate identity is a net in a Banach algebra or ring (generally without an identity) that acts as a substitute for an identity element.

A right approximate identity in a Banach algebra A is a net ${\displaystyle \{\,e_{\lambda }\colon \lambda \in \Lambda \,\}}$ such that for every element a of A, ${\displaystyle \lim _{\lambda \in \Lambda }\lVert ae_{\lambda }-a\rVert =0.}$ Similarly, a left approximate identity in a Banach algebra A is a net ${\displaystyle \{\,e_{\lambda }\colon \lambda \in \Lambda \,\}}$ such that for every element a of A, ${\displaystyle \lim _{\lambda \in \Lambda }\lVert e_{\lambda }a-a\rVert =0.}$ An approximate identity is a net which is both a right approximate identity and a left approximate identity.

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