Is it possible to visualize the theorems and proofs in functional analysis?

Question

I am an MSc.second year student and just started studying functional analysis.I understand the proofs of the theorems in this topic but since I cannot visualize those things,so I forget the proofs easily as I do not get what is going on actually and just understand the proof step by step.So,my question is: Is it possible to visualize the theorems and proofs in functional analysis?

First difficulty I face is due to not knowing the standard normed spaces very well.For example,I do not know how $L_p[0,1],1\leq p<\infty$ and $L_\infty[0,1]$ spaces behave unlike I know for $\mathbb R^n$ and $\mathbb C^n$.These spaces seem to much abstract to work with.Also finding counterexamples in the spaces $\ell_p,\ell_\infty$ seem to be difficult.Compared to that working with $(C[0,1],\|.\|_\infty)$ is quite easy.

Also it is difficult to work with infinite dimensional normed spaces because we are used to finite dimensional normed spaces and try to visualize the subspaces by lines or planes$($as we have in $\mathbb R^3)$,but things are not so simple in infinite dimensional cases.So,I fail to create a mental picture of a normed subspace of a normed space.This troubles me a lot.

Then comes theorems like uniform boundedness principle and open mapping theorem.As constructing examples in functional analysis is not so easy,I face problems in finding intuition behind the proofs we are taught in class.

So,I am struggling a lot in functional analysis.Can someone tell me a way out?I mean can someone tell me a proper way to study functional analysis and visualize these things?

Answer 1

Well, I think you are doing well. You are trying to compare with finite dimensional spaces, and you are getting the differences. This is all you can do, together with "be patient". Eventually, you will get a mental picture for this new spaces.

I only can add what Von Neumann once said:

"Young man, in mathematics you don't understand things. You just get used to them."

Edit: For some results, it has been useful for me doing the opposite. An element of $f\in L_2([0,1])$ is a function, with a graph. So, in orther to compare with a finite dimensional Hilbert space, I change the visualization of $\mathbb{R}^3$ to a kind of "bar diagram" with three columns. In the same way that functions are graphs, vector in $\mathbb{R}^3$ are also this particular case of discrete "graphs".

Answer 2

There are a lot of examples in a book called "Introductory functional analysis with applications" by Erwin Kreyszig, such as Legendre polynomials, spline interpolation, quantum mechanics, approximations using Chebyschev polynomials, differential equations, numerical integration, etc.