Schauder Basis with Finite Blaschke Products

We construct a Schauder basis for the space $Hol(\mathbb{D})$, the space of holomorphic functions on the closed unit disk, consisting entirely of finite Blaschke products. The expansion coefficients are given explicitly. Our result remains valid when $Hol(\mathbb{D})$ is equipped with a broader class of norms satisfying natural structural conditions. These conditions are satisfied by norms of classical function spaces such as the Hardy spaces $H^p$ ($1\le p\le \mathrm{\infty }$), the weighted Bergman spaces $A_{\alpha }^p$ ($1\le p\le \mathrm{\infty }$, $\alpha >-1$), and BMOA. We also establish the optimality of this framework by proving that such a basis cannot exist in larger spaces, such as the Hardy space $H^p$ and the disc algebra $A(\mathbb{D})$.