Let \(G\) be an arbitrary bounded region in the complex plane, \(\text{Hol}(G)\) a space of functions holomorphic in \(G\) with topology of uniform convergence on compact sets in \(G\), and \(H^ \infty(G)\) be an algebra of bounded functions with the usual sup norm. Suppose that \(Q(G)\) is a topological space of functions holomorphic in \(G\) with topology \(\tau\) such that \(H^ \infty(G)\subseteq Q(G)\subseteq \text{Hol}(G)\) and that \(\tau\) is an ``intermediate topology'' in the following sense: The topology induced from \(Q(G)\) onto \(H^ \infty(G)\) is weaker than the topology generated by the sup norm and the topology \(\tau\) is stronger than that induced from \(\text{Hol}(G)\) onto \(Q(G)\). Examples of such spaces include Hardy spaces, Bergman spaces, BMOA spaces, and many others. The author establishes some general properties of the orthonormal systems of polynomials in the Hilbert spaces with intermediate topology. These include analogs of the Bergman kernel function and the relationship of this function to certain extremal problems. Results are obtained for the special case of Hardy spaces, with some hypotheses on the domain \(G\). The properties of orthonormal polynomials in arbitrary Hilbert spaces with intermediate topology are also studied.