Concerning the index notation: Nielsen & Chuang are considering arbitrary finite sums of the form $$\sum_{i\in I} a_i \left| v_i \right\rangle \otimes \left| w_i \right\rangle, \qquad a_i~\in~\mathbb{C},$$ in the tensor product $V\otimes W$, where the index set $I$ is finite: $|I|<\infty$. There is no assumption of, say, linear independence of $$ \left| v_i \right\rangle, \qquad i\in I.$$ Nor is there an assumption of linear independence of $$ \left| w_i \right\rangle, \qquad i\in I.$$ In particular, it is not necessary to assume that the vector spaces $V$ and $W$ have the same dimension.

Concerning the index notation: Nielsen & Chuang are considering arbitrary finite sums  $$\sum_{i\in I} a_i \left| v_i \right\rangle \otimes \left| w_i \right\rangle, \qquad a_i~\in~\mathbb{C},$$ in the tensor product $V\otimes W$, where the index set $I$ is arbitrary but finite: $|I|<\infty$. There is no assumption of, say, linear independence of $$ \left| v_i \right\rangle, \qquad i\in I.$$ Nor is there an assumption of linear independence of $$ \left| w_i \right\rangle, \qquad i\in I.$$ In particular, it is not necessary to assume that the vector spaces $V$ and $W$ have the same dimension.

FWIW, Nielsen & Chuang are considering finite sums of the form $$\sum_{i\in I} a_i \left| v_i \right\rangle \otimes \left| w_i \right\rangle, \qquad a_i~\in~\mathbb{C},$$ in the tensor product $V\otimes W$, where the index set $I$ is finite: $|I|<\infty$. There is no assumption of, say, linear independence of $$ \left| v_i \right\rangle, \qquad i\in I.$$ Nor is there an assumption of linear independence of $$ \left| w_i \right\rangle, \qquad i\in I.$$ In particular, it is not necessary to assume that the vector spaces $V$ and $W$ have the same dimension.

Concerning the index notation: Nielsen & Chuang are considering arbitrary finite sums of the form $$\sum_{i\in I} a_i \left| v_i \right\rangle \otimes \left| w_i \right\rangle, \qquad a_i~\in~\mathbb{C},$$ in the tensor product $V\otimes W$, where the index set $I$ is finite: $|I|<\infty$. There is no assumption of, say, linear independence of $$ \left| v_i \right\rangle, \qquad i\in I.$$ Nor is there an assumption of linear independence of $$ \left| w_i \right\rangle, \qquad i\in I.$$ In particular, it is not necessary to assume that the vector spaces $V$ and $W$ have the same dimension.