Definition of tensor product (2-dimension): Let $H_1$ and $H_2$ is Hilbert space. For $\psi\in H_1$, $\phi\in H_2$, a map $\psi\otimes\phi:H_1\times H_2\to \mathbb{C}$ is defined by $\psi\otimes \phi(u,v)=\langle \psi,u\rangle_{H_1}\langle \phi,\ v\rangle_{H_2}$. Then, \begin{equation}H_1\hat{\bigotimes}H_2=\left\{\sum_{i=1}^n\psi_i\otimes \phi_i: \psi_i\in H_1, \phi_i\in H_2\right\}\end{equation} is a vector space. This is called algebraic tensor product. \begin{equation} \left\langle\sum_{i=1}^n\psi_i\otimes \phi_i\ ,\ \sum_{j=1}^m \psi_j'\otimes \phi_j'\right\rangle=\sum_{i=1}^n\sum_{j=1}^m\langle \psi_i, \psi_j'\rangle_{H_1}\langle \phi_i,\phi_j'\rangle_{H_2} \end{equation} is innner product. Completion of $H_1\hat{\bigotimes}H_2$ for this inner product is tensor product. This is denoded by $H_1\bigotimes H_2$.
My text say that $\bigotimes^n\mathbb{C}\simeq\mathbb{C}$ (clearly, $\mathbb{C}$ is Hilbert space) by natural isomorphism. I cannot understand this. It is not $\mathbb{C}^n$ ? I'd appreciate it if you could help.