It is well - known that a point - wise multiplication operation can be defined on some function spaces.
For example, $C([0,1])$, the vector space of a real - valued (say) continuous functions $f$ defined on the unit - interval $[0,1] \subset \mathbb{R}$, admits such a multiplication, defined as \begin{equation} (fg)(x) := f(x)g(x), \quad f,g \in C([0,1]) \end{equation}
Now, for function spaces whose elements are defined on a countable set, say \begin{equation} \mathbb{R}^k = \{ x: \mathbb{N}_k \to \mathbb{R}, \quad n \mapsto x_n, \quad n = 1, \dots, k \} \quad (k \in \mathbb{N}) \end{equation} or \begin{equation} \mathbb{R}^\infty = \{ x: \mathbb{N} \to \mathbb{R}, \quad n \mapsto x_n, \quad n = 1,2 \dots \,\} \quad \end{equation}
I wonder why this is not done analogously. So, define pointwise multiplication by \begin{equation} (xy)(n) := x_ny_n \end{equation}
For example, in the "column - notation", two elements of $\mathbb{R}^2$, $x = (x_1,x_2)$ and $y = (y_1,y_2)$ would then have the product $ xy = (x_1y_1, x_2y_2)$.
I realize there must be something that makes this obvious operation utterly useless, for otherwise, it would be used commonly. Any hints as to why this attempt to define pointwise multiplication on these functions spaces is uninteresting would be great !
From what I understand so far (unfortunately far to little), I wonder whether the cardinality of the domain (uncountable in the case $[0,1]$, countable in the cases $\mathbb{N}_k = \{1,\dots k\}$ and $\mathbb{N}$) makes a crucial difference, whether there is some algebraic property that breaks down, or whether it is something completely different ?
Thanks for your feedback and help!