Bohr's theorem [\textit{H. Bohr}, Proc. Lond. Math. Soc. (2) 13, 1-5 (1914; JFM 44.0289.01)] states that given an analytic function in the unit disk, \(f(z) = \sum_k c_k z^k\), such that \(|f(z)|0\) such that \(\|f\|_{K_1} \leq |f|_{K_2} \). The main abstract result of the paper is that if \(H(M)\) has a basis \(\{ \varphi_n, n \geq 0\}\) such that \(\varphi_0 =1\) (which is a necessary condition for the Bohr property) and that there exists \(z_0 \in M\) such that \(\varphi_n (z_0) =0\), \( n\geq 1\), then \(H(M)\) has the Bohr property. In another section, the authors consider the case where the space \(H\) is a Hilbert space of analytic functions on a bounded domain \(D \subset \mathbb C^n\), and \(\{ \varphi_n\), \(n \geq 0\}\) is an orthogonal basis. Furthermore, the Hilbert norm is an \(L^2\) norm with respect to a Borel measure \(\mu\), and point evaluations are continuous. Suppose further that \(\mu\) is representing for a point \(z_0 \in D\). Then a Bohr property takes place iff there exist an open set \(U \ni z_0\), a constant \(C>0\), and a compact set \(K\) such that \(\|f\|_U \leq C |f|_K\), for all bounded holomorphic \(f\). An application of this is given to show that certain doubly orthogonal bases enjoy the Bohr property.