Let $\mathcal{L}$ be a cocycle on $X$ and $\lambda \in \mathbb{R}$. A family $E \colon \Omega \to \mathcal{G}_k X$ of $k$-dimensional subspaces is a fast space of multiplicity $k$ at rate $\lambda$ when it is measurable and equivariant ($\mathcal{L}_\omega E(\omega) = E(\sigma\omega)$ almost surely), and almost surely every nonzero vector of $E(\omega)$ has growth rate $\lambda$ and $\tfrac1n \log g(\mathcal{L}^{(n)}_\omega, E(\omega)) \to \lambda$.