# Visualizing $L_2$ or an infinite dimensional Hilbert space

I need to explain abstract objects in an infinite dimensional Hilbert space. What is the best way to visualize it for an engineering audience? Does anybody have a good example?

• I honestly do not see what is it you expect! :-) Just make pictures on the blackboard and say «and pretend this is happening in an infinite dimensional Hilbert space». – Mariano Suárez-Álvarez Sep 3 '12 at 18:23
• imagine n dimensions and let n = infinity – binn Sep 3 '12 at 18:25
• I'm unsure that one should try to visualize $L_2$. You can have intuition for something without a (very very approximate) visual representation. – Alex Becker Sep 3 '12 at 18:33
• @AlexBecker Good point. I know it's going to be an approximation. But I wanted to carry as much intuition as possible. – Memming Sep 6 '12 at 1:08

## 2 Answers

I wouldn't go for a visualization, especially for engineers. I think a much better approach is to speak about an infinite number of degrees of freedom. For engineers one could use Fourier-series: A signal $f:[0,T]\to\mathbb{C}$ of finite energy (which means $L^2$ but you do not need to tell this at the first place) has an infinite (in fact countable, but this also does not matter) number of degrees of freedom which can be expressed but the Fourier-coefficients. Then you can move on and explain how to calculate these coefficients and you see that it amounts to the use of an "inner product" and there you are.

Alternatively, you could also use "square summable sequences" but I think they are not that well suited (at least, I do not have a good "real world model" at hand).

All you can really visualize or draw is a two- or three-dimensional picture. Fortunately, two- and three-dimensional pictures often suffice. You just have to be able to look at a line and pretend that it is an infinite-dimensional subspace.