Necessity and utility of infinite dimensional spaces

Question

I have a group of similar questions about infinite dimensional spaces. I think these are a little discursive and trivial, but I can't find any answer.

-Why is the concept of infinite dimensional spaces so important and useful?

-From which field did the concept arise and for which necessity?

-Is this concept in some cases related to describing objects with a non finite number of degrees of freedom? If so, does this connection hold for every case?

Answer 1

There are entire books answering these questions. Here is a nutshell answer to your three questions.

1. The concept is important since it arises in applications, primarily (and the origin of the research into it) in quantum mechanics as well as in differential equations. It is useful since the theory of infinite dimensional spaces is very well behaved and thus offers practical techniques for solving hard problems.

2. Similar to the above. The field arose from formulations in physics around the turn of the 20th century that used spaces of functions.

3. Yes. No.