Definition 36.12.2 (Approximate Identity).label Let $A$ be a Banach algebra and $\angles{e_\beta}_{\beta \in B}\subset A$ be a net, then $\angles{e_\beta}_{\beta \in B}$ is an approximate identity of $A$ if:

1. (1)

   For each $\beta \in B$, $\norm{e_\beta}_{A}\le 1$.

2. (2)

   For every $x \in A$, $e_{\beta} x \to x$ and $x e_{\beta} \to x$.