> infinite linear combinations of orthogonal functions
I'm not sure it's quite accurate to say that these can't even begin to be visualised. The theory of Fourier series means that someone picturing a 'reasonable' function on the circle has already begun the endeavour.
You're right. I thought, even while I was writing that, that it was a weak example. Though I had Banach spaces in mind, not fourier series, and I have always had trouble with them.
Still. I felt I needed three examples. "Pick something not from set theory," I said to myself. I couldn't come up with a strong example. Maybe that's telling.
Unmeasurable sets, though. When I try to visualize one, I see the letter E because that's what we called it in the constructive proof.
You're in good company. Supposedly, someone once asked J G Thompson (a very eminent group theorist, winner of the Fields Medal, the Wolf Prize and the Abel Prize) what mental picture he had in his head when thinking about a difficult group theory problem. "A big black letter G", he said.
At James Arthur's 60th birthday conference, he said something like the following (roughly paraphrased): "First …" and, unfortunately, I don't remember what first was (but I agree with Dove (http://news.ycombinator.com/item?id=1385842) that there should be 3 :-) ).
"Next, I studied the work of Langlands, and the group was called GL_2. Then, I studied the work of Harish-Chandra, and the group was called G."
Exotic differential structures (brief explanation: consider 4-dimensional Euclidean space, R^4; you can think of its topological structure as being determined by the way you calculate distances between points; this actually gives you more than just topological structure because you can do things like differentiate functions on the space. Well, there are different distance functions that are equivalent topologically but not differentially. The same is true for lots of other spaces, in dimensions much higher than 4.)
Huge finite groups like the "Monster".
The Galois group of Qbar over Q. (Brief explanation: Q is the rational numbers. Qbar is the set of all "algebraic numbers", i.e. roots of polynomials with integer coefficients. The Galois group consists of all "field automorphisms of Qbar", which means all functions from Qbar to itself that don't mess with arithmetic: f(x+y)=f(x)+f(y), f(xy)=f(x)f(y), etc. It's an important object in number theory.)
The very infinite-dimensional Hilbert space in which the wavefunction of the universe lives.
I'm not sure it's quite accurate to say that these can't even begin to be visualised. The theory of Fourier series means that someone picturing a 'reasonable' function on the circle has already begun the endeavour.