The Brownian motion has the following (Levy-Ciesielski?) construction via Hilbert space isomorphisms:

Let $\{ Z_i \}_{i \in \mathbb{Z}}$ be i.i.d. $N(0,1)$ random variables defined on $(\Omega, \mathcal{F}, P)$.

Let $\{ \phi_i \}_{i \in \mathbb{Z}}$ be an orthonormal basis for $L^2[0, 1]$.

The map $\phi_i \mapsto Z_i $ extends to an isometry $B$ from $L^2[0, 1]$ to the subspace of $L^2(\Omega)$ generated by $\{ Z_i \}_{i \in \mathbb{Z}}$.

The process defined by $B_t(\omega) = B(1_{[0,t]})(\omega)$ is a version of Brownian motion, and $\int_0 ^1 \phi_i dB_t = Z_i$ in the Ito sense.

Does this extend in some way to the fractional Brownian motion?