“Visualizing” Mathematical Objects - Tips & Tricks

Question

It has been a while since I am kind of stuck with my skills concerning the visualization of mathematical objects.

Here there is the problem.

First of all, let me point out that I am completely self-taught. In other words, this actually means that this site is the only possibility I have to speak mathematics, which is a bit like trying to learn japanese by rarely pronouncing some utterance to a random native speaker hoping not to sound too idiotic.
In second place, I am trying to move towards rather abstract things, and –from time to time– I do have the feeling I have a decent grasp of what is going on. However…

There are some objects that I simply don’t know how to approach!

I think the problem can be rephrased in terms of extensive vs. intensive definition of a given mathematical object. I always tend to look for an extensive definition, but –when you start to deal with real analysis, topology or measure theory– those kind of definitions, that quite literally show how an object looks like, simply start to rarify.

Thus, when I find a definition of –let’s say– $\ell^p$, or $\mathbb{N}^{\mathbb{N}}$, or Borel Sets, then I really fail to see what is going on. Maybe I can actually manipulate the symbols, and even get a “proof” or something that could almost look like a proof, but still, in most of the cases I have no idea what is going on.
[Small aside: this is actually interesting. As long as definitions and objects are sort of trivial, there are "pictures" or advice to visualize objects, but when you get to the level of those objects, it is assumed that the reader/learner somewhere has got the skills to visualize those structures.]

If you want an analogy, it is a bit like being in a completely dark room, with various objects and the explicit task to make them fit perfectly. I can do it (from time to time), but this does not imply that I see what the single objects where at the beginning, and how they look like now that they are assambled. Said so, here it comes the questions.

How do you actually perceive or visualize the mathematical objects?
How did your teachers/professors/supervisors taught you to visualize them?
How do you teach your students to visualize mathematical objects?
What are the “tricks”?

There is something else that should be added, and that I think is related to the fact that I am self-taught. A lot of books of advanced maths simply discard “pictures”, even if (I suppose) maybe the authors were implicitly referring to them when they started to learn the topics. But here there is the point of not-being self taught: there is somebody who gives you this “tricks” (if you don’t like the word, we can use "upaya" instead) to build, and then erase every trace of the use of those hints.

Thank you for your time and for any feedback that could come!

PS: For those who are wondering, I added the real-analysis and measure-theory tags, because I would love to actually see the objects I was referring in the text, and to get tips and hints on how to do it.

EDIT: After the nice feedback from user86418, I would like to point out something. The idea behind the question is not to head to the psychology of mathematics, or in other words how each user see things in her/his own mathematical world. Actually, the idea is to look for the general tips that are shown on the whiteboard (quite literally!) to visualize objects. An example could be (if I remember correctly history of math) the Argan plane: Gauss was the first to get the idea, but he erased any mention to it, and people were kind of stuck to visualize properly complex numbers for a while!

Answer 1

$\newcommand{\Reals}{\mathbf{R}}$My advisor would draw a rectangle with a line underneath and say, "Let $P$ be a principal bundle...." I, by contrast, would draw circle bundles over curved base spaces. (Each picture was useful for certain types of question. The point is, my advisor didn't really teach me how to visualize.)

As for the type of visualization in the question: I'm not an analyst, but here's how I visualize $\ell^{p}$.

First, I think of the space $\Reals^{\omega}$ of real sequences as "spanned" by countably many orthogonal lines. (I use "spanned" in an informal geometric sense, not in the sense of linear algebra. Visually, these lines float against a black background, are blue-ish, and fade to transparent as the index increases.) The space $\Reals^{\infty}$ of finite sequences looks like the "truncated" subspace where nearly-transparent axes of unspecified index are "clipped" to the origin. This conveys more-or-less the same "feel" as a plane in $\Reals^{3}$.

Alternatively, I think of vertical lines in the plane, one line over each positive integer, so that an element of $\Reals^{\omega}$ is a collection of "beads", one on each line.

Now for $\ell^{p}$: I think of the graphs $y = \pm x^{-1/p}$, and imagine sequences (collections of beads) where the beads lie between the graphs. Translated into "orthogonal axes", this picture becomes a product of segments of decreasing length; the larger $p$ is, the more slowly the length decreases as the index grows (and the blue density fades). This is literally incorrect for a couple of reasons, but I find it's a compelling image.