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 $e_k$).
Question Since $X_t$ is written in terms of basis $\{e_k\}$ of $L^2[a,b]$, is it correct to assume $X_t: (\Omega,\mathcal A,P)\to L^2[0,1]$, that is, $X_t$ is in the (infinite dimensional) family of functions $L^2[0,1]$? What about $X$?
Comments In my case, $X=\{X_t, t\in [0,1]\}$ is the Wiener process defined on $(\Omega,\mathcal A, P)$ into some infinite dimensional space $F$. I'll apply the Karhunen-Loève expansion for this Wiener process. But I need to make it explict which set $F$ is.
Update It seems that the Karhunen-Loève Theorem works with another point of view: the random functions in $X=\{X_t, t\in[0,1]\}$ are viewed as $X_t\in L^2(\Omega,\mathcal A, P)$ (i.e., the equivalent class of random vables with finite second moment) which is a separable Hilbert space. Hence the stochastic process is given by $[0,1] \to L^2(\Omega,\mathcal A,P)$ with $t\mapsto X_t$, and not as $\Omega \to F$ with $\omega\mapsto X_t(\omega)$ as I was trying to do before.
Can someone give me directions? I'm not an expert in this field.