The Karhunen-Loève Theorem is concerned about a continuous second-order process $X=\{X_t, t\in [a,b]\}$, defined on a probability space $(\Omega,\mathcal A, P)$.
The Theorem allows us to apply the following expansion:
$$X_t=\Sigma_{k=1}^\infty Z_ke_k(t), \quad t\in [a,b],$$ where $\{e_k\}$ are the eigenfunctions of the (Hilbert-Schmidt integral operator) linear operator $T_{K_X}:L^2[a,b]\to L^2[a,b]$ and $\{Z_k\}$ are zero mean pairwise orthogonal random variables with corresponding variance $\lambda_k$ (the eigenvalues associated to $e_k$).
It is known that $\{e_k\}$ form an orthonormal system in $L^2[a,b]$. I infer that the space spanned by $\{e_k\}$, named $V$, is a subspace of $L^2[a,b]$. If I equip $L^2[a,b]$ with the standard inner product $<,>$, then $(L^2[a,b],<,>)$ is a separable Hilbert space.
Question Is $(V,<,>)$ again a separable Hilbert space?