Continuous Dimensional Spaces I was wondering, is there such a thing as a 'continuous dimensional space' in mathematics?
For instance, consider $\mathbb{R}^2$. If this can be thought of as two $\mathbb{R}$'s, does there exist something of the form $\mathbb{R}^\mathbb{R}$ ? But not for a single superscript real number, but rather the whole set at once.
What does such a space mean for real numbers? 
I'm having a hard time trying to express my thoughts in words. That might be because I'm saying something ridiculous (very likely). Anyways, could someone please explain?
 A: In set theory, the notation $A^B$ (where $A,B$ are sets) stands for the set of all functions from $A$ to $B$. We are used to think of $\mathbb{R}^2$ as consisting of tuples $(a,b)$ of real numbers but we might as well think of $\mathbb{R}^2$ as the set $\mathbb{R}^{\{ 1, 2\}}$. The latter set consists of functions $f \colon \{ 1,2 \} \rightarrow \mathbb{R}$ from a set $\{ 1, 2 \}$ with two elements (whose members I decided to conveniently call $1,2$ but specific elements don't really matter) to $\mathbb{R}$. The translation between tuples and functions goes like this:
$$ f \mapsto (f(1),f(2)), \\ (a,b) \mapsto f(x) = \begin{cases} a & x = 1, \\
b & x = 2. \end{cases} $$
so we can think of $f(1)$ as giving us the "first coordinate" of a tuple and of $f(2)$ as giving us the second coordinate. 
This might look a bit strange in the beginning but it allows you to interpret $\mathbb{R}^\mathbb{R}$ as the set of functions from $\mathbb{R}$ to $\mathbb{R}$. To specify a point in $\mathbb{R}^7$ we need to give seven real numbers or a function from a set with seven elements to $\mathbb{R}$ and to specify a point in $\mathbb{R}^{\mathbb{R}}$ we need to give "$\mathbb{R}$ real numbers" or, in other words, a function from $\mathbb{R}$ to $\mathbb{R}$. This space has the structure of a vector space which is not finite dimensional although I wouldn't call it a "continuous dimensional space".
A: Yes. In fact, in set theory, $A^B$ means "the set of functions from $B$ to $A$". This aligns nicely with the idea of the space $\mathbb{R}^n$: it's the set of functions from an $n$-element set (say $\{1,2,\dotsc,n\}$) to $\mathbb{R}$. This is the same as saying that the elements are $n$-tuples $(r_1,r_2,\dotsc,r_n)$.
There's no difficulty in extending the abstract definition to, say, $\mathbb{N}$: now you have a countable set, and so the elements are tuples of the form $(r_1,r_2,\dotsc)$, where the list doesn't terminate. You may recognise this: it's a sequence.
To come to your question, $\mathbb{R}^{\mathbb{R}}$ can therefore be thought of as "functions from $\mathbb{R}$ to $\mathbb{R}$", i.e. the set of $f:\mathbb{R} \to \mathbb{R}$.
When $V$ is a vector space, $V^n$ is also a vector space by making operations act componentwise (i.e. $(x+y)_k = x_k+y_k$ for each $1 \leqslant k \leqslant n$ and so on). This still works if we take $V^{\mathbb{R}}$, except we say that addition and scalar multiplication act pointwise: in particular, you might know already that "functions from $\mathbb{R}$ to $\mathbb{R}$" is a vector space when we set $(f+g)(x) = f(x)+g(x)$ and $(\lambda f)(x) = \lambda \cdot f(x)$.
