Let $\mathcal{L}$ be a cocycle on a Banach space $X$, and $B_c(\omega)$ the set of vectors with a backward history at $\omega$ decaying at rate $c$. Then

• each $B_c(\omega)$ is a linear subspace of $X$, and $B_c(\omega) \subseteq B_{c'}(\omega)$ for $c' \le c$;
• if $X$ is separable with its Borel $\sigma$-algebra and $E : \Omega \to \mathcal{G}_m X$ is a measurable family with $\mathcal{L}_\omega E(\omega) = E(\sigma\omega)$ almost surely, growing at rate at least $c$ (for every $b < c$, $g(\mathcal{L}^{(n)}_\omega, E(\omega)) \ge e^{bn}$ for all large $n$, almost surely), then $E(\omega) \subseteq B_c(\omega)$ almost surely.