With $\{ |e\rangle_j \}_{j=1}^{dim. \mathcal{H}_A}$ for $\mathcal{H}_A$ and $\{|f\rangle_j \}_{j=1}^{dim. \mathcal{H}_B}$ for $\mathcal{H}_B$, the product state reads \begin{equation} |u\rangle \otimes |v\rangle = \left(\sum\limits_{j} a_j |e\rangle_j \right) \otimes \left(\sum\limits_{k} b_k |f_k\rangle \right) = \sum\limits_{j,k} a_j b_k |e_j\rangle \otimes |f_k\rangle, \end{equation} where $a_j$, $b_k$ are complex number.
What is the expalaination of the following statement: If the vectors $|u\rangle$ and $|v\rangle$ do not belong to the respective orthonormal bases then there are at least two coefficients $a_j$ and similarly at least two coefficients $b_k$. From this, we conclude that the state is entangled.
Edit: Here is the original version from the book The Mathematical Language of Quantum Theory by Mário Ziman and Teiko Heinosaari (Chapter 6).