I read in the notes of Stochastic Processes that there is a construction of Wiener Process (knowing that $Cov(W_s, W_t)=min(s,t)$ ) which going like this:
consider operator $Q$ on $C([0,1])$ $$Qf(t)= \int_0^1 min(s,t) f(s) ds $$ find the functions $f_k$ such, that $$Qf_k= \lambda_k f_k$$ then the Wiener Process has a form: $$W(t)= \sum_{k=0}^\infty a_k X_k f_k$$
in the notes, there are also explicit forms of $f_k$ and $a_k$. My question is why this algorithm works? Does someone know the proof?
reference - part of italian notes to Stochastic Processes course: http://en.file-upload.net/download-11221924/psZ.pdf.html