Definition 32.4.1 (Ideal).label Let $A$ be a Banach algebra and $I \subset A$, then $I$ is a left ideal if:

1. (1)

   $I$ is a subspace of $A$.

2. (2)

   For each $x \in A$, $aI \subset I$.

and $I$ is a two-sided ideal if in addition to the above,

1. (3)

   For each $x \in A$, $Ia \subset I$.

In this document, the ”sidedness” of ideals are omitted. Within each block, the statement should hold as long as the interpretation is consistent.