I'm reading Nielsen and Chuang, _Quantum Computation and Quantum Information._ On p. 73 it is introducing inner products and tensor products. So it says the following:

> The inner product on the spaces $V$ and $W$ can be used to define a natural inner product on $V \otimes W$. Define

> $$\big(\sum_i a_i \left | v_i \right\rangle \otimes \left | w_i \right\rangle, \sum_j b_j \left | v_j' \right\rangle \otimes \left | w_j' \right\rangle\big) \equiv \sum_{ij} a_i^*b_j \left\langle v_i | v_j' \right\rangle \left\langle w_i | w_j' \right\rangle.\tag{2.49} $$

This only makes sense if $V$ and $W$ have the same dimension. Say, $V$ is 2-dimensional and $W$ is 3-dimensional. What does $\left | v_3 \right\rangle$ mean? I would appreciate some clarification.