Approximate unit/identity in Banach and or C*-algebra

Question

In Banach algebras and C*-algebras, the concept of approximate unit (or approximate identity) is defined as nets, rather than sequences. My question is, since the underlying space is Banach and the convergence is norm-convergence, why do we need to use nets, rather than sequences?

Answer 1

From Blackadar's "Operator Algebras - Theory of $C^*$-Algebras and von Neumann Algebras'':

II.4.2.4 Proposition. A $C^*$-algebra contains a strictly positive element if and only if it has a countable approximate unit. (Such a $C^*$-algebra is called $σ$-unital.)

A positive element $h$ in a $C^*$-algebra $A$ is said to be strictly positive iff $φ(h)>0$, for every state $φ$ on $A$.