“Visualizing” Mathematical Objects - Tips & Tricks

Solution 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.