What does the vector space $\mathbb{R}^{\mathbb{R}}$ look like?

Question

I can imagine $\mathbb{R}^{\mathbb{N}}$. For instance, the set of real series is part of this space, as is any infinite (but discrete numbered) tuple of reals.

But how can I imagine $\mathbb{R}^{\mathbb{Q}}$, or even $\mathbb{R}^{\mathbb{R}}$?

Answer 1

To see what $\mathbb{R}^\mathbb{R}$ looks like, it helps to have a more general notion. For any two sets $A, B$, we define $A^B = \{f: B \to A\}$, i.e. the set of all functions from $B$ to $A$. The motivation for this definition is that we naturally want $A^B$ to consist of "$B$-tuples" of elements of $A$. How does one come up with such a thing? Easy! Just assign an element of $A$ to each element of $B$. That is, define a function $f: B \to A$.

Thus, if one wants to picture $\mathbb{R}^\mathbb{R}$, one simply thinks of the set of all functions $f: \mathbb{R} \to \mathbb{R}$. Likewise, $\mathbb{R}^\mathbb{Q} = \{f: \mathbb{Q} \to \mathbb{R}\}$. Note that since $\mathbb{Q}$ and $\mathbb{N}$ have the same cardinality, the sets $\mathbb{R}^\mathbb{N}$ and $\mathbb{R}^\mathbb{Q}$ are really "the same" in some sense (in particular, they are isomorphic as vector spaces).

Answer 2

It's very hard to put a geometry on infinite dimensional spaces. It's even harder when we want to think about them topologically, which is what you are trying to do here.

See, as vector spaces $\Bbb{R^N}$ and $\Bbb{R^Q}$ and even $\ell_2$ or $\ell_\infty$ are all the same. These are all vector spaces over $\Bbb R$ whose dimension is $2^{\aleph_0}$. Where does the difference come in, then? It comes in the form of the natural topology we associate with the vector space. And this makes a good sense, too. The reason these are not really difference is that every vector space has a basis, which is a consequence of the axiom of choice. And without the axiom of choice it is consistent that these spaces are different from one another (well, not $\Bbb{R^N}$ and $\Bbb{R^Q}$, but these two from $\ell_2$ and $\ell_\infty$).

And the axiom of choice is the black box of the imagination. It guarantees the existence of some objects which are intangible to us, we cannot really imagine them in a clear way. Which means that we can't quite see why the vector spaces are isomorphic (as vector spaces, not as normed spaces or whatever). But the mathematics shows that they are, and that's fine.

But what's $\Bbb{R^R}$? This set is unimaginably larger than $\Bbb R$ in its cardinality. So it is different from those other vector spaces that I have mentioned on the merit of its size alone.

I can't tell you how to imagine these objects. I can, to some extent, see them in my head, but those are things I could have never described with words. Instead, at a certain size, the structure that we can put on objects needs to be dealt with on a smaller scale when approaching them. That means that picking up a few objects and trying to size them into a finite dimensional subspace. And we work a lot with the definitions.

People put too much emphasis on "seeing things". In mathematics you don't "see things" at first. You have to shed the limitations of your physical experience, by working closely with the definitions. After you've understood the definitions, then you start to have a picture in your head. And that picture is yours, and yours alone, because you can't transfer it back to the real world. But you can transfer the definitions, and using them communicate your ideas and understanding with other mathematicians that hold their own image in their own heads.

Answer 3

I don't know if this helps to visualize what $\mathbb{R}^ \mathbb{R}$ looks like, but here is how I visualize an element of that vector space: I picture a number line, and attached to each point, in addition to the label identifying the point, there is a second label containing another number. Of course the points on the number line are dense, so the labels are all on top of one another and you can't see them clearly, but in my visualization you see more and more labels as you "zoom in" on a small interval.

A "doubly-labeled" number line like this is the closest I can get to imagining an "uncountable tuple". So then $\mathbb{R}^ \mathbb{R}$ is the set of all such doubly-labeled number-lines.