We all know what $\mathbb{R}^n$ means. But I came across this statement about $\mathbb{R}^\mathbb{N}$ in a note that says $\mathbb{R}^\mathbb{N}$ is a vector space under pointwise operations has no natural norm. The note provides no further explanation about it.
My questions are:
1. What is $\mathbb{R}^\mathbb{N}$? How do we call it? Is it a Euclidean vector space?
2. What does pointwise operations mean? And why does it not have a natural norm?
I tried to google $\mathbb{R}^\mathbb{N}$ but did not seem to find a good explanation.
Thank you!
Generally, $A^B=\{f\colon B\to A\}$ is the set of functions from $B$ to $A$. So, $\mathbb R^{\mathbb N}$ is the set of functions $f\colon \mathbb N \to \mathbb R$. Since a function $f\colon \mathbb N \to \mathbb R$ is essentially the same as a sequence $(a_n)$, we call $\mathbb R ^{\mathbb N}$ the set of sequences in $\mathbb R$. It is not a Euclidean vector space.
By pointwise operation we mean that we define the structure by restricting attention to each coordinate separately. E.g., $\alpha \cdot (a_n)=(\alpha \cdot a_n)$. The fact that this gives rise to a vector space is a triviality that you should verify to yourself. Finally, the lack of a natural norm is the (somewhat vague and imprecise) statement that the standard norm on $\mathbb R^n$, namely $\langle u,v\rangle=\sum_k^nu_kv_k $ does not extend to $\mathbb R^{\mathbb N}$, since it requires replacing $n$ by $\infty $, but there is no reason for the series to converge in general.