The following has been lectured
Defn (Product Spaces) For any $n \in \mathbb{N}$ and metric spaces $\left(M_k, d_k\right)$ for $k \in\{1, \ldots, n\}$, we define the product space $$ \left(\bigoplus_{k=1}^n M_k\right)_p=M_1 \oplus_p \cdots \oplus_p M_n=\left(M_1 \times \cdots \times M_n, d_p\right) . $$ Example (Product spaces) We have that $\mathbb{R} \oplus_1 \mathbb{R}=\ell_1^2$. Remark $\mathbb{R} \oplus \mathbb{R} \oplus \mathbb{R}$ doesn't make sense since we have not defined the associativity of the $\oplus$ operator. The two choices yield different metric spaces.
Here $$ d_p (x,y) = \left( \sum_{i=1}^{n} \ (d_i (x_i, y_i))^{p} \right)^{1/p} $$
I am struggling to understand both the definition and the example given. In the definition, does it tell us that $M_1 \times \cdots \times M_n$ is the "Set" and $d_p$ is the metric? Moreover, if so, why doesn't $\mathbb{R} \oplus \mathbb{R} \oplus \mathbb{R}$ make sense? Should it not be $\ell _1 ^3$?
If anybody could clarify that for me I would greatly appreciate it.
