Annihilators in infinite-dimensional Grassmann–Banach algebras

A family of infinite-dimensional Grassmann–Banach algebras over a complete normalized field $K$ is considered. It is proved that any element $G$ of the family is an associative supercommutative Banach superalgebra over $K$, i. e. $G=G_0\oplus G_1$ with the zero annihilators $G_0^\perp=G_1^\perp=(G_1^{(k)})^\perp=\{0\}$, $k\geqslant2$.