As there already are answers addressing the notation issue, I will just add some examples and intuition.
Consider a function $f : \{1,2,3,\ldots,n\} \to \mathbb{R}$. However, there is a bijection between a set of such functions and the $\mathbb{R}^n$. Take an $n$-dimensional vector $v \in \mathbb{R}^n$ such that $v = \langle v_1, v_2, v_3, \ldots, v_n \rangle$ and $v_k = f(k)$, and you could see that $\Phi(f) = v$ could be an example. Of course, if you have a point in $u \in \mathbb{R}^n$, then you could easily reconstruct the function: $g(k) = u_k$. The fact that the space of the functions is denoted as $\mathbb{R}^{\{1,2,\ldots,n\}}$ is not a coincidence.
This example can be generalized, e.g. take $f : \mathbb{N} \to \mathbb{R}$. Every such function is a point in the space of infinite sequences of real numbers, e.g. $(a_n)_{n \in \mathbb{N}}$ where $a_n = f(n)$. On the other hand, the space of such sequence can be thought of as an infinite Cartesian product of $\mathbb{R}$, and indeed it is somtimes denoted as $\mathbb{R}^\infty$.
We can go with this even further and further, e.g. $g : [a,b] \to \mathbb{R}$ is a point in space $\mathbb{R}^{[a,b]}$ that can be understood as infinite Cartesian product, but with uncountable number of coordinates it ranges over.
Please note, that with the above in mind, the sums (i.e. $\sum_{k=1}^n \cdot$) and series (i.e. $\sum_{k=1}^\infty \cdot$) are nothing else than the integrals (i.e. $\int_{\Omega}\cdot$) over discrete spaces. This is also one of the reasons why people tend to think of integrals as generalized sums.
Hope this helps ;-)