a) Let $V_{1}, V_{2}, \ldots, V_{k} $ be vector spaces with bases $ B_{1}, \ldots, B_{k}$. Specify a basis of the Cartesian product $ V=V_{1} \times V_{2} \times \cdots \times V_{k} $.
b) Specify a basis of $ \mathbb{R}^{(\mathrm{N})}$.
Note: Show that your answer is actually a basis in each case.
Here is my solution approach:
(It would be great if someone could have a look at my solution for part a) and help me with part b))
a)
$ v=\sum \limits_{j=1}^{n} \lambda_{j}\left(b_{j}\right) \sum \limits_{i=1}^{m} \mu_{i}\left(\beta_{i}\right)\left(b_{j}, \beta_{i}\right)$ of $V_{1}, V_{2}$ because
$v_{1} \in V_{1} \quad v_{1}=\sum \limits_{j=1}^{n} \lambda_{j}\left(b_{j}\right) b_{j}$
$v_{2} \in V_{2} \quad v_{2}=\sum \limits_{i=1}^{m} \mu_{i}\left(\beta_{i}\right) \beta_{i}$
$\vdots$
$ v_{k} \in V_{k} \quad v_{k}=\underbrace{\left.\sum \limits_{l=1}^{r} \phi_{l}\left(\gamma_{l}\right) \gamma_{l}\right)}_{\text {basis of } V_{k}}$
hence:
$ v=\left(\sum \limits_{j=1}^{n} \Phi_{j}\left(b_{j}\right) b_{j}, \ldots, \sum \limits_{l=1}^{r} \Phi_{l}\left(\gamma_{l}\right) \gamma_{l}\right)=\sum \limits_{\gamma=1}^{n} \cdot \ldots \cdot \sum \limits_{l=1}^{r}\left(\lambda_{j}\left(b_{j}\right) b_{j}, \ldots, \Phi_{l}\left(\gamma_{l}\right) \gamma_{l}\right)$
$=\underbrace{\sum \limits_{\sigma=1} \lambda_{j}\left(b_{j}\right) \cdot \ldots \cdot \sum \limits_{l=1}^{r} \Phi_{l}\left(j_{l}\right) \cdot\left(b_{j}, \ldots, \gamma_{l}\right)}_{\text{Cross product of bases of the vector spaces}}$
b)
I know that it's a $\mathbb{N}$-tuple of the bases, but how do I write that down?