Understanding Fractional Pdes

Suppose we have an orthonormal basis given by $\{ \phi_k \}_{k \geq 1}$ of $L^2(\Omega)$. This implies that we can write for all $w \in L^2(\Omega)$ as $$ w = \sum_{k=1}^{\infty} w_k \phi_k \, \, , \,\, \text{where} \,\, w_k = (w_k, \phi_k)_{L^2(\Omega)} = \int_{\Omega} w \phi_k \, dx \,\, .